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UC-NRLF 77fl . \ - JVo, Division Shelf- Received OK THK of . HISTORY OF THE THEORIES OF ATTRACTION AND THE FIGURE OF THE EARTH. VOLUME I. Get admirable Ouvrage [Newton's Principia] contient les gennes de toutes les grandes decouvertes qui ont 6t< faites depuis sur le systeme du monde: 1'liistoire de leur developpement par les successeurs de ce grand ge"ometre serait a la fois le plus utile commentaire de son Ouvrage, et le meilleur guide pour arriver a de nouvelles ddcouvertes. LAPLACE. Connaissance des Terns pour Van 1823. A HISTORY OF THE MATHEMATICAL THEORIES OF ATTRACTION AND THE FIGURE OF THE EARTH, FROM THE TIME OF NEWTON TO THAT OF LAPLACE. BY I.^TODHUNTER, M.A., F.R.S. IN TWO VOLUMES. VOLUME I. Hon&on : MACMILLAN AND CO 1873. [All Rights reserved.] - \ T CTamtrftge : PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. PREFACE. IN the volumes now offered to students I have written the history of an important branch of science in the manner in which I formerly treated the Calculus of Variations and the mathe- matical theory of Probability ; and in the present work, as in those, I undertake a task hitherto unattempted. For although much has been published on the History of Astronomy, yet the progress of the mathematical development of the principle of Attraction has been left almost untouched. The last of the six volumes which constitute the great work of Delambre is devoted to the Astronomy of the eighteenth century ; but the Astronomy discussed is almost entirely that of observation, and the investi- gations of the eminent mathematicians who contributed to fill up the outline traced by Newton are scarcely noticed. There are indeed interesting and valuable works in which the results obtained by theory are stated in popular language for the benefit of general readers ; such is the well-known history by Bailly in French, with its continuation by Voiron; and in English we have various excellent productions of the same kind, especially Narrien's Historical Account of the Origin and Progress of Astro- nomy, and Grant's History of Physical Astronomy. But the object of these works is quite distinct from that which I have kept in view in my contributions to scientific history. I desire not merely to record the results which may have been obtained but to trace the analysis which led to those results, to estimate its value, and to discriminate between its failure and its success, its error and its truth. So far as I know the only example of a mathematical treatise bearing on the history of Physical Astronomy is Gautier's Essai Historique sur le probleme des trois corps : but as this treats of the Lunar and Planetary Theories, omitting the Figure of the Bodies, it has nothing in common with the present work. In the fifth volume of the Mtcanique Celeste Laplace arranges the whole subject of Physical Astronomy in six divisions, and gives brief sketches of the progress of the theory of all : in every case sound knowledge practically begins with Newton. Laplace's first division is devoted to the Figure and Rotation of the Earth ; and this has suggested to me the subject of the present work. I T. M. A. b VI PREFACE. undertake accordingly to trace the history of the Theories of Attraction and of the Figure of the Earth from Newton to Laplace. The two subjects are necessarily associated in origin, and have been historically always united ; they are discussed together by Laplace in the second volume of his great work. I have confined myself to a single division of the wide subject of Physical Astro- nomy, for the extent and difficulty of the whole might deter even a professional cultivator of the science ; and the numerous un- finished fragments of works intended to bear on the Me'canique Celeste furnish an impressive warning against the rashness of any extravagant design. I will now give an outline of the plan of my work. The first Chapter is necessarily occupied with Newton, the founder of Phy- sical Astronomy. The power revealed in all his efforts is nowhere more conspicuous than in his treatment of our two subjects. In the theory of attraction, among other important results, he shewed that the attraction of a spherical shell on an external particle is the same as if the shell were collected at its centre, and that the attraction on an internal particle is zero. These two pro- positions constitute a complete theory of the attraction of a sphere in which the density varies as the distance from the centre. More- over the result with respect to an internal particle was extended by Newton to the case in which the bounding surfaces of the shell are similar, similarly situated, and concentric ellipsoids of revo- lution. Newton originated the idea of investigating the Figure of the Earth on the supposition that it might be treated as a homogeneous fluid rotating with uniform angular velocity. He assumed as a postulate that there could be relative equilibrium in such a case if the form were that of an oblate ellipsoid of revolution ; and he determined the ratio of the axes and the law of variation of gravity at the surface. The investigation, though not free from imperfection, is a rare example of success in the first discussion of a most difficult problem, and constitutes an enduring monument to the surpassing ability of its author. The second Chapter is devoted to Huygens. To him we owe the important condition of fluid equilibrium, that the result- ant force at any point of the free surface must be normal to the surface at that point ; and this has indirectly promoted the knowledge of our subject. But Huygens never accepted the great principle of the mutual attraction of particles of matter ; and thus he contributed explicitly only the solution of a theoretical pro- blem, namely the investigation of the form of the surface of rotating fluid under the action of a force always directed to a fixed point. PREFACE. VH The third Chapter treats of various miscellaneous investiga- tions connected with the subject in the course of one generation after the publication of the Principia. No real addition was made to Newton's theoretical results, while the measurements of arcs of the meridian in France led the Cassinis to adopt the hypothesis that the form of the Earth was not oblate but oblong. The fourth Chapter relates to Maupertuis. He wrote various memoirs, among which were two in the form of commentaries on Newton's theories of Attraction and the Figure of the Earth. These theories were rendered more accessible by the translation from their original geometrical expression into the familiar analy- tical language of the epoch. By adhering to Newton's conclu- sions Maupertuis must have contributed much to maintain the truth among his countrymen, in opposition to the errors recom- mended by the authority of Des Cartes and the Cassinis. The important postulate assumed by Newton was first con- sidered by Stirling, a mathematician of great power: the fifth Chapter shews that he obtained, at least implicitly, an approxi- mate demonstration of the required result. In the sixth Chapter an account is given of various memoirs by Clairaut which preceded the publication of his important work on the Figure of the Earth. Clairaut explicitly demonstrated the truth of Newton's postulate approximately. He also gave the theorem, called Clairaut's, theorem, which establishes a connection between the ellipticity of the earth and the coefficient of the term expressing the increase of gravity in passing from the equator to the pole. The seventh Chapter narrates briefly the circumstances of the measurement of an arc of the meridian in Lapland. I have undertaken to develop the progress of the Mathematical Theories of Attraction and of the Figure of the Earth ; but I do not profess to include the practical operations conducive to our knowledge of the exact dimensions of the Earth. These consist mainly of obser- vations of pendulums, and measurements of arcs ; and an account of them drawn from the original sources would form an interest- ing and instructive work. But the more difficult matters to which I have devoted the present volumes have furnished ample employment without any serious divergence into the department of practical application. I have therefore limited myself to short notices of the earlier pendulum experiments, and of the two great measurements in Lapland and Peru ; these measurements deserve some attention on account of their historical interest and their decisive testimony to the oblate form of the Earth. The eighth Chapter treats of various miscellaneous investi- gations between 1721 and 1740. Desaguliers maintained, with 62 Vlll PREFACE. a zeal not uniformly discreet, the oblate form against the Cassinian hypothesis ; on the other hand, the measurements in France were still held to be in favour of that hypothesis. Towards the end of the period the Academy of Paris proposed the Tides as the subject of a Prize Essay; and this led to the important researches of Maclaurin. The ninth Chapter is devoted to Maclaurin. He completely solved the problem of the attraction of an ellipsoid of revo- lution on an internal or superficial particle ; and his method and results admitted of obvious extension to the case of an ellipsoid not of revolution. The extent to which he proceeded for the case of an external particle requires to be stated with accuracy, in order to correct errors of opposite kinds which are current. The most general result yet attained may be stated thus : the potentials of two confocal ellipsoids at a given point ex- ternal to both are as their masses. This theorem was first es- tablished by Laplace, but Maclaurin demonstrated it for the par- ticular case in which the external point is on the prolongation of an axis of the ellipsoids. In the theory of the Figure of the Earth, Maclaurin' s main achievement was an exact demonstra- tion of Newton's postulate, of which hitherto only approximate investigations had been given. In the tenth Chapter the contributions of Thomas Simpson are noticed. This eminent mathematician explicitly shewed that if the angular velocity of rotation exceeds a certain value, the oblatum is not a possible form of relative equilibrium for a fluid mass ; and it followed implicitly from his results that for any value of the angular velocity less than the limit, more than one figure for relative equilibrium would exist. Simpson also gave a remarkable investigation of the attraction at the surface of a very extensive class of nearly spherical bodies. The eleventh Chapter consists of an analysis of the celebrated work by Clairaut. The first part of the work treats on the principles of fluid equilibrium ; here Clairaut far surpassed his predecessors in extent and accuracy, and left the theory in the form which it still retains, with the single exception of the im- provement effected by Euler, who introduced the notion of the pressure at any point of the fluid, together with the appropriate symbol by which it is denoted. The second part of the work treats on the Figure of the Earth. For the case of a homogeneous fluid Clairaut closely followed Maclaurin. The case of a hetero- geneous fluid had been hitherto practically untouched, and Clairaut invented for it a beautiful process which has remained substan- tially unchanged to the present time ; the chief result is a certain equation connecting the ellipticity of the strata with their density, PREFACE. ix which appears in two forms: these I have called " respectively Clairaut's primary equation, and Clairaut's derived equation. The twelfth Chapter narrates briefly the circumstances of the measurement of an arc of the meridian in Peru. I have care- fully examined the extensive literature, much of which is con- troversial, arising from this memorable expedition ; and by means of exact references I have afforded assistance to any student who wishes to render himself familiar with all the circumstances. The thirteenth Chapter is devoted to the earlier half of the writings of D'Alembert which bear on our subjects. They are extensive in amount, and may have served indirectly to diffuse the interest in such investigations which the writer must have felt himself; but on account of errors in principle and inaccuracy of detail their direct value is small. In various attempts which D'Alembert made to criticise the work of Clairaut he was I believe almost uniformly wrong, so far as regards the Figure of the Earth, and barely right on some unimportant points of Hydrostatics. It is stated in the life of D'Alembert published in the Biographical Dictionary of the Society for the Diffusion of Useful Knowledge that " He and Clairaut were rivals, and no work of either appeared without rinding a severe critic in the other; but D'Alembert, the more cautious and profound of the two, was generally on the right side of the question:..." The judgment is pronounced by a most eminent authority to which I usually bow with reverence ; but so far as the subjects of the present work extend, I should venture to reverse it. The fourteenth Chapter is devoted principally to Boscovich, whose writings furnish elementary accounts of the most important results which had been obtained up to their date. I have also given a brief notice of the poem by Stay, for which Boscovich supplied notes and supplementary dissertations. The fifteenth Chapter treats of various miscellaneous investi- gations between the years 1741 and 1760. It includes a brief notice of a Prize Essay on the Figure of the Earth, published by Clairaut, some years after his treatise. The sixteenth Chapter is occupied with the later half of the writings of D'Alembert. The general character is the same as of the earlier half; the investigations themselves are disfigured by serious errors, but they serve to suggest interesting and important matter. The works of Frisi are noticed in the seventeenth Chapter: they resemble those of Boscovich in the fact that they served to teach the subject rather than to promote its progress. The eighteenth Chapter treats of various miscellaneous in- vestigations between the years 1761 and 1780. The first three X PREFACE. of Laplace's memoirs belong to this period, but for convenience the consideration of them is postponed. The Chapter includes an ac- count of a memoir by Lagrange in which he proceeded by analysis to the point Maclaurin had reached by geometry. The operations carried on at Schehallien for ascertaining the density of the Earth are noticed, and references are supplied to the subsequent labours on the same subject. Here the first volume ends, which contains the history of our subjects during the century which followed the publication of Newton's Principia. The nineteenth Chapter takes the first three memoirs of Laplace. The principal object of these memoirs may be said to be the solution of a problem which is an extension of Newton's postulate. Newton assumed that an oblatum was a possible form of relative equilibrium for rotating fluid ; the present problem is to shew that an oblatum is the only possible form, at least under certain restrictions. I call the problem Legendre's, because he was the first who solved it with tolerable success. D'Alembert attempted the investigation, but failed. Laplace did not solve the problem completely; but he shewed that for a very large class of nearly spherical figures, the relative equilibrium was im- possible. He also obtained the expression for the law of gravity which would hold universally. The twentieth Chapter is devoted to a memoir which is con- spicuous in the history of the Theory of Attraction, namely the earliest of Legendre's. The limit reached by Maclaurin is now for the first time left behind ; Legendre shews that the theorem with respect to confocal ellipsoids is true for any position of the ex- ternal point when the ellipsoids are solids of revolution. Legendre introduces here the memorable expressions, hitherto unknown, which are now usually called Laplace s coefficients ; and also, at the suggestion of Laplace, the function now called the Potential func- tion takes its place in the subject. The twenty-first Chapter brings before us a scarce treatise by Laplace, and gives an analysis of that half of it which relates to Attraction and the Figure of the Earth. Here was published for the first time, the demonstration of the theorem relating to the action of confocal ellipsoids at an external point which I call by Laplace's name. The subjects of the Attraction of Ellipsoids and of the homogeneous Figure of the Earth appear in this treatise in nearly the same form as in the Mecanique Celeste. The twenty-second Chapter relates to Legendre's second me- moir. Here Legendre solves the problem which I call by his name. He assumes that the fluid is in the form of a figure of revolution, and that it does not deviate widely from the spherical form. PREFACE. XI The twenty-third Chapter notices Laplace's fourth, fifth, and sixth memoirs. The fourth and fifth memoirs contain the theory of the attraction of spheroids, and the theory of Laplace's functions, in the form they assume in the M&anique Celeste. The sixth memoir relates to Saturn's ring. The twenty-fourth Chapter is devoted to Legendre's third memoir. The object of this memoir is to demonstrate Laplace's theorem, respecting confocal ellipsoids by a more direct process than Laplace himself had employed. Legendre does demonstrate the theorem, without expanding his expressions in series, but the process is excessively long and complicated. The twenty-fifth Chapter analyses Legendre's fourth memoir. Here we have a great development of Clairaut's process for the case of heterogeneous fluid. A general equation is obtained analogous to Clairaut's primary equation; and from this it is shewn that the strata must be ellipsoidal. The twenty-sixth Chapter is devoted to Laplace's seventh me- moir. This contains some numerical discussion of the lengths of degrees, and of the lengths of the seconds pendulum; there is also a theory of the heterogeneous figure of the Earth, which sub- stantially agrees with that in Legendre's fourth memoir. The twenty-seventh Chapter treats of miscellaneous investiga- tions between the years 1781 and 1800. Among other matters we have here to notice Cousin's Introduction to the study of Physical Astronomy, a memoir by Lagrange, and a memoir by Trembley ; the last is of the same unsatisfactory character as various memoirs by the same writer which I have examined in my History of the Mathematical Theory of Probability. The twenty-eighth Chapter gives an account of the first two volumes of the Me'canique Celeste, so far as they relate to our subjects. Laplace in effect reproduced with small change the last four of his seven memoirs ; and the result is a treatise not yet superseded. The twenty-ninth Chapter traces the history of investigation with respect to Laplace's Theorem. Ivory, Legendre, Gauss and Rodrigues all gave complete discussions of the attraction of ellip- soids; while Biot and Plana also commented on parts of the theory. The method of Ivory is the simplest of all, and has obtained a permanent position in our elementary works ; insomuch that it is usual to speak of Ivory s theorem, although the more correct phrase would be Ivory s demonstration of Laplace s theorem. The thirtieth Chapter treats on an equation which Laplace seems to have regarded with peculiar favour, and which occurs often in his writings. The equation however did not satisfy Ivory, and he criticised it with severity. The result of the discussion xil PREFACE". may be said to have established the accuracy of Laplace's equation when used, as he himself used it, with due caution. But at the same time the objects which Laplace sought by the aid of his equation are now generally obtained without it ; so that practically the equation is at present rarely employed. The thirty-first Chapter elucidates the partial differential equation for the symbol which denotes the potential function. Laplace had originally assumed that a certain equation held both for an external particle, and for a component particle of the body considered ; but Poisson shewed that the two cases required different forms of the equation. The thirty-second Chapter discusses a method which Laplace gave for solving Legendre's problem, with the objection brought against it by Liouville, and the treatment which Poisson substi- tuted in place of Laplace's. The thirty-third Chapter passes in review various memoirs which Laplace published during the first quarter of the present century. The thirty-fourth Chapter is devoted to that part of the fifth volume of the Mecanique Celeste which relates to our subjects ; it consists chiefly of a republication of the memoirs noticed in the thirty-third Chapter. Strictly speaking the period of history which I proposed to describe closes here ; but it seemed convenient to include within my range all the writings of three mathematicians who had already been prominent in my work, and who may be naturally associated with their predecessors, especially with Laplace. These writers are Poisson, Ivory and Plana. The thirty-fifth Chapter contains an account of all Poisson's contributions which had not been previously examined. The most important of these are an elaborate memoir on the Attraction of Spheroids, and a memoir giving a new investigation of Laplace's theorem respecting confocal ellipsoids. The thirty-sixth Chapter gives a brief sketch of the numerous articles and memoirs which Ivory produced, mainly in support of opinions of his own w r hich were both peculiar and erroneous. The great promise which his early success held out was not followed by any corresponding merit in the essays of his later years. The thirty-seventh Chapter is devoted to Plana, who wrote several papers chiefly in the form of comments on Lagrange, Legendre and Laplace. The last Chapter treats of various miscellaneous investigations during the first quarter of the present century. It is by accident the history finishes with a paragraph relating to Bowditch; but on account of his moral and intellectual eminence, and of his unselfish devotion to science, the name of one of the most dis- PREFACE. Xill tinguished mathematicians beyond the Atlantic may justly close a roll which commences with that of Newton. The period of time which I have traversed will be found to correspond with some accuracy to a distinct boundary line in the subject. The labours of more recent date present to us many in- dications of what may be more appropriately called new methods rather than mere developments of those already discussed. Among them we may mention the investigations respecting the Potential by Green and Gauss, and the numerous researches on the attrac- tion of Ellipsoids by Chasles ; all these writers will occupy conspicuous places in any future record of the subjects. Sir John Herschel spoke of my History of Probability as embracing the series of the Pleiocene analysts in distinction from the Post-Pleiocene ; aad the illustration might be similarly applied in the present case. Such then is the outline of the history which the present volumes contain. The principles on which I have executed my task are the same as those adopted in my former works ; and I may refer especially to the preface to my History of Probability for an account of them. I will only state here that I have not thought it necessary to preserve the exact notation of the original authors ; that notation frequently varies much in various places, and it is really advantageous for the sake of brevity and clearness to use the same symbols throughout. For example the ratio of the centrifugal force at the equator to the gravity there is denoted in some English books by the letter m ; Clairaut uses <f> ; D'Alembert in the sixth volume of his Opuscules Math&matiques uses o> ; Laplace in the Mecanique Celeste, Vol. v. page 7, uses <>, and in Vol. v. page 23 he uses a<f>. For the ratio of the centri- fugal force at the equator to the attraction there, which is very approximately the same thing as the preceding ratio, the letter j is used throughout the present work. I have been very sparing in the introduction of new terms, for this practice seems carried to an embarrassing extent in some modern mathematical works. I have however found it necessary to have short designations for two things which occur perpetually in these investigations. The body formed by the revolution of an ellipse round its minor axis I call an oblatum, and the body formed by the revolution of an ellipse round its major axis I call an 06- longum. In English books the former has usually been called an oblate spheroid ; and the latter a prolate spheroid. Something is gained in conciseness by using one word instead of two for a name which is frequently required ; but the chief reason of the change arises from the fact that the word spheroid has been much used in a different sense, namely to denote a body which differs but little from a sphere. It would be very convenient if this sense XIV PREFACE. of the word spheroid could be so established as to render superfluous the formal enunciation of the condition of resemblance to a sphere. Perhaps the use of a word to express a form only approximately determined is felt to be somewhat unlike the ordinary precision of mathematical language ; and this may account for the frequent repetition of the condition even after it has been explicitly adopted. Moreover the great French writers have often employed the word spheroid in a sense so wide as to render it practically equivalent to body; an example will be found in the title of a memoir by Poisson on page 388 of the second volume. I have found it convenient to give a name to a certain ratio which is of importance in our subject, namely the ratio of the difference of gravity at the equator and at the pole to gravity at the equator. This ratio is one of the elements connected by Clairaut's theorem, and I have accordingly called it Clairaut's fraction. There is one term, perhaps the most objectionable of all that have become permanent in mixed mathematics, which is used throughout the work, namely centrifugal force. It is with great reluctance that I have felt myself constrained to yield to uni- versal authority and to employ language which experience shews to be most perplexing and misleading. The well-trained student will however have learned that the so-called centrifugal force is a fiction ; the simple fact is that a dynamical problem relating to a body which is rotating uniformly, can be reduced to a statical problem by supposing the rotation to cease and a certain force to be introduced. This History assumes on the part of the reader some elementary acquaintance with the subjects on which it treats. For the Theory of Attractions the Chapter in my work on Statics, to which I have occasionally referred, will be sufficient. For the Figure of the Earth the student may consult three well-known English treatises, namely one in Airy's Mathematical Tracts, one in O'Brien's Mathematical Tracts, and Pratt's Chapter on the subject in his Mechanical Philosophy, afterwards enlarged and published separately in a Treatise on Attractions, Laplace's Functions and the Figure of the Earth : Pratt's Treatise is the most comprehen- sive of these English treatises, and the easiest to procure. An interesting work was published at Paris in 1865, entitled Traitt El&mentaire de Mecanique Celeste. Par H. Resal. About a third of this volume is devoted to our subjects ; and it gives a very instructive account of them : but the extreme inaccuracy of the printing is a serious diminution of the value of the work. The mathematical expressions which are called Laplace's coefficients and Laplaces functions play a very important part in PREFACE. XV the higher investigations of our subjects. The treatises of O'Brien, Pratt, and Resal, which have just been cited contain a sufficient account of these expressions for elementary purposes. The student who wishes to become intimately acquainted with them will have recourse to the work by Heine which is named on page 24 of the second volume ; this is an admirable volume enriched with numerous references to the original authorities. It may be naturally expected that a person who has devoted much time to the study of the history of science will feel disposed to attribute considerable value to the pursuit. The interest which attaches to the struggle of the human mind with serious difficul- ties, to its gradual progress and final triumph, may be at least as great as that which is excited by an account of the vicissitudes of civil history. An acquaintance with the origin and the course of any science will often give great assistance in the comprehension of its present state, and may even point out the most promising direc- tion for future efforts. Moreover a familiarity with what has been already accomplished or attempted in any subject is conducive to a wise economy of labour ; for it may often prevent a writer from investigating afresh what has been already settled, or it may warn him by the failure of his predecessors, that he should not too lightly undertake a labour of well-recognised difficulty. The opinions of Laplace and Arago, which are quoted in my title-pages, are justly entitled to great weight on these points. That the subjects here treated historically are of no common importance and influence may be easily seen. A knowledge of the figure and dimensions of the Earth is the basis of all the numerical results of Astronomy, and therefore of the greatest practical value. Moreover the researches into the theories of Attraction and of the Figure of the Earth have been fertile in yielding new resources for mathematicians ; it will be sufficient to point to the Transfor- mation of Multiple Integrals, the theory of the Potential, and the elaborate doctrine of Laplace's functions, which have all sprung up in the cultivation of this field of Physical Astronomy. Humboldt has drawn attention to this circumstance in his Cosmos; the fol- lowing passage occurs on pages 156 and 157 of the fifth edition of Sabine's translation of the first Volume : " Except the investi- gations concerning the parallax of the fixed stars, which led to the discovery of aberration and nutation, the history of science presents no problem in which the object obtained, the knowledge of the mean compression of the Earth, and the certainty that its figure is not a regular one, is so far surpassed in importance by the incidental gain which, in the course of its long and arduous pursuit, has accrued in the general cultivation and advancement of mathematical and astronomical knowledge." XVI PEEFACE. It may appear that some apology is due for the extent to which the work has grown ; this must be found in the extent and intricacy of the materials which had to be analysed. Indeed Ivory, who devoted much attention to the subject of the Figure of the Earth, asserts that it has been attended with greater difficulty and has occasioned a greater number of memoirs than any other branch of the system of the world. I have had some trouble in keeping within the limits of two volumes, and have been com- pelled to omit many developments which I should gladly have printed. I have also published separately various papers which have grown out of my historical studies ; to these I refer in the appropriate places, but it may be convenient to give a list of them here. They are the following : On Jacobi's Theorem respecting the relative equilibrium of a revolving ellipsoid of fluid, and on Ivory's discussion of the Theorem. Proceedings of the Royal Society, Vol. xix. Note relating to the Attraction of Spheroids. Proceedings of the Royal Society, Vol. xx. Note on an erroneous extension of Jacobi's Theorem. Pro- ceedings of the Royal Society, Vol. xxi. On the Arc of the Meridian measured in Lapland. Trans- actions of the Cambridge Philosophical Society, Vol. XII. On the equation which determines the form of the strata in Legendre's and Laplace's theory of the Figure of the Earth. Transactions of the Cambridge Philosophical Society, Vol. xn. On the Proposition 38 of the Third Book of TSewtoria Prinoipia. Monthly Notices of the Royal Astronomical Society, Vol. xxxn. On the Arc of the Meridian measured in South Africa. Monthly Notices of the Royal Astronomical Society, Vol. xxxm. The account which is given of the memoirs and treatises will be found ample enough in most cases to supply all that a student will ever want to read of them ; but this does not apply to the Mtcanique Celeste, which I desire to illustrate not to super- sede. In other words all that I say relative to that great work is intended as a commentary for the use of those who are consulting the original. I have usually cited it by sections, but in some cases, which occur almost exclusively in the fifth volume, I have for greater distinctness cited it by pages. The pages meant are those of Laplace's own edition; but the student who uses the national edition will be able to adjust the references by observing that in the fifth volume the 85 pages with which we are concerned correspond to 103 pages in the national edition. It is well known that Laplace does not give any specific PREFACE. XV11 references to the labours of his predecessors and contemporaries ; in his great treatises on Physical Astronomy and Probability he embodied with his own results much that he derived from others, and as these treatises have become the standards of authority for the subjects to which they relate, it has followed that with un- critical readers Laplace has not unfrequently obtained credit for what was not distinctively his own production. A student of the course of science will often discover that important investigations which first came under his notice in the works of Laplace, are really due to other mathematicians ; and by a natural reaction the conjecture will arise that further research will lead to the restitution of much more to the rightful owners ; and thus there may be a recoil from an undue admiration to a suspicious depre- ciation. But a complete evolution of the history will restore the reputation of Laplace to its just eminence. The advance of mathematical science is on the whole remarkably gradual, for with the single exception of Newton there is very little exhibition of great and sudden developments ; but the possessions of one generation are received, augmented, and transmitted by the next. It may be confidently maintained that no single person has contributed more to the general stock than Laplace. In the life of Laplace in the English Cyclopaedia, which we may safety attribute to the late Professor De Morgan, there are some valuable remarks suggested by the want of specific informa- tion in the writings of Laplace as to what was done by himself and what was done by others; and it is stated that no one has yet supplied the deficiency. With respect to Laplace it is said: "Had he consulted his own glory, he would have taken care always to note exactly that part of his own work in which he had a forerunner; and it is not until this shall have been w r ell and precisely done, that his labours will receive their proper apprecia- tion." In the present history and in that of Probability I have gone over a third part of the collected mathematical works of Laplace ; and to that extent the evidence of his great power and achievements is I hope fully and fairly manifested. I have not hesitated to criticise all that has come before me ; and there is scarcely any memoir or treatise of importance left without the suggestion of corrections or additions. I cannot venture to hope that 1 have uniformly escaped without any obscurity or error. My readers will I trust excuse such blemishes, arising partly from the nature of the task and partly from the circumstance that only such leisure could be found for it as remained amidst continuous occupation in elementary teaching and writing. The work has thus furnished ample employment for seven years of labour, with the exception of a necessary digres- XV111 PREFACE. sion in order to explain and illustrate some peculiarities in the Calculus of Variations. It was perhaps rash for a mere volunteer to undertake so extensive a task ; but in spite of the imperfec- tions with which it may have been accomplished, I am willing to hope that the result will be a permanent addition to the literature of Physical Astronomy. It is not from any desire to challenge comparisons with illus- trious men, but merely to justify my estimate of the labour involved, that I venture to quote the following opinion expressed by the late Professor James Forbes in his Review of the Progress of Mathematical and Physical Science, and to extend its applica- tion from pure to mixed mathematics : " Specimens of what a history of pure mathematics would be, and must be, are to be found in the able ' Reports ' of Dr Peacock and Mr Leslie Ellis, in the Transactions of the British Association for 1833, and 1846. A glance at these profound and very technical essays will shew the impossibility of a popular mode of treatment, while the dif- ficulty and labour of producing such summaries may be argued from their exceeding rarity in this or any other language." I have to record my great obligations to the Rev. J. Sephton, Head Master of the Liverpool Institute, formerly Fellow of St John's College, for his most valuable assistance in conducting the work through the Press. To the Syndics of the University Press I am indebted for their liberality in defraying the expenses of the printing. I. TODHUNTER. ST JOHN'S COLLEGE, CAMBRIDGE, July, 1873. CONTENTS. VOLUME I. PAGE CHAPTER I. NEWTON 1 Publication of the Principia, i. Attractions, 2. Spherical shell on an in- ternal particle, 3. Spherical shell on an external particle, 4. Lemma, 6. Zone of an indefinitely thin spherical shell, 7. Sphere on an external particle, 8. Sphere on an internal particle, 10. Solid of revolution on a particle at any point of the axis, 13. Infinite plane lamina, 15. Jesuits' edition, 16. Density of the Earth, 17. Figure of the Earth, 18. Polar and equatorial canals of fluid, 19. Approximate estimate of attrac- tions, 20. Oblatum and oblongum, 21. "Remarks on the approximate estimate, 22. Equilibrium of the canals, 23. Attraction, gravity, weight, 25. Newton's value of the ellipticity, 26. Jupiter's figure, 29. Influence of the Sun's heat, 30. Measured lengths of degrees, 32. Weight resolved along the radius, 33. Increment of gravity, 34. Pendulums, 36. Newton's error, 37. Table of the lengths of a degree, 39. Pendulums, 40. The Cassinian hypothesis untrue, 43. Summary of results, 44. Laplace's opinion, 45. Halley's opinion, 46. CHAPTER II. HUYGENS * 28 Publication of the Discourse, 47. Date of composition, 49. Vortex, 50. Air pump, and weight in a mine, 51. Pendulum at Cayenne, 52. Principle in Hydrostatics, 53. Value of the ellipticity, 54. Equation, 55. Result extended, 56. Remarks on the Principia, 59. The Sun's distance from the Earth, 61. Resisting medium, 62. Huygens's problem, 64. Mistake as to priority, 65. CHAPTER III. MISCELLANEOUS INVESTIGATIONS UP TO THE YEAR 1720 37 Norwood, 68. Pendulums, 69. The Arabian measure, 70. Norwood, 71. Halley, 72. Burnet, Whiston, and Keill, 73. Incidental mistakes, 75. XX CONTENTS. PACK KeilPs error, 76. Keill and Halley, 77. Keill and Bentley, 78. Keill and Whiston, 79. Pendulums, 80. D. Cassini adopts Keill's error, 81. Snell, 82. De La Hire, 83. D. Gregory, 84. Keill, 85. Quotations from Keill and Arago, 87. Numerical result, 89. Pendulums, 90. Freind, 91. J. Cassini, 92. Hermann, 93. Hermann's problem, 95. Quotations from Hermann and Boscovich, 97. J. Cassini, 99. French arc, 100. Mairan, 109. Invents a law of attraction, 1 13. Summary of results, 115. CHAPTER IV. MAUPERTUIS 63 Saturn's ring, 117. First problem, 118. Second problem, 119. Figure dea Astres, 122. A difficulty, 124 5866725. Variable stars and nebulae, 127. On the laws of attraction, 128. Incidental statements, 130. Figure de la Terre, 131. Figures des Corps Celestes, 132. Hydrostatical princi- ples of Newton and Huygens, 133. Mistake, 137. View of an obscure passage in Newton, 138. Figure de la Terre, 140, 141, 142. Examen de'sinte'resse', 143. CHAPTER V. STIRLING 77 Newton's postulate, 151. Sir J. Lubbock, 152. Resultant action at the sur- face of an oblatum, 153. Pendulum, 155. Remarks on the merits of Clairaut and Stirling, 156. Theory compared with fact, 157. Estimates of Stirling, 158. CHAPTER VI. CLAIRAUT 83 Geodesic curve, 160. Proposition in solid Geometry, 161. Arcs of meri- dian, 162. Newton's postulate, 163. Approximate attraction of an obla- tum at the pole, 165. Fluid with variable density, 167. Unsatisfactory with respect to Hydrostatics, 170. Clairaut's Fraction, and Clairaut'a Theorem, 171. Huygens's problem, 173. Hydrostatical principles, 174. Geodesic curve, 177. CHAPTER VII. ARC C OF THE MERIDIAN MEASURED IN LAPLAND 93 Peruvian company, 178. Lapland company, 179. Maupertuis's book, 181. Outhier's book, 182. Selection of places, 184. Difference of latitude, 185. Measurement of the base, 186. Difference of latitude redetermined, 189. Hardships, 192. Incidental matters, 194. Ellipticity, 196. Svanberg, 197. Celsius, 198. Reference for further information, 199. CHAPTER VIII. MISCELLANEOUS INVESTIGATIONS BE- TWEEN THE YEARS 1721 AND 1740 . . .103 Desaguliers, 200. Criticises the conclusions of J. Cassini, 201. Admits Keill's error, 202. Criticises Mairan 's memoir, 203. Considers the French arc, 204. CONTENTS. XXI PAGK An experiment, 205. Incidental matters, 206. Poleni, 209. De la Croyere, 210. J. Cassini, 211. Godin, 213. La Condamine, 214. J. Cassini, 215. Pendulum, 217. Manfredi, 218. Bouguer on Hydro- statics, -219. J. Cassiui, 220. Prize Essay by John Bernoulli, 211. Modes of determining the form of the Earth, 222, 223. Cassini de Thury, 224. Pendulums, 225. Cassini de Thury, 226. Bouguer, 227. Delisle, 228. Euler, 229. D. Bernoulli's Hydrodynamica, 230. Cassini de Thury, 231. The Tides, 232. D. Bernoulli's Essay, 233. Euler's Essay, 234. Arc between Paris and Amiens, 235. French arc, 236. Remark on the error in Picard's base, 238. Jesuits' edition of the Principia, 239! Winsheim, 240. CHAPTER IX. MACLAURIN 133 Treatise of Fluxions, 241. Attractions, 242. Ellipsoid of revolution, 244. Newton's Postulate, 245. Extends Newton's Hydrostatical principle, 246. Value of gravity, 247. Level surfaces, 248. Newton's Postulate, 249. Attraction of a sphere, 251. Of an ellipsoid of revolution at the equator and pole, 252. Reference to Newton, Cotes, and Stirling, 253. Attrac- tion of confocal ellipsoids, 254. Maclaurin's demonstration, 255. Con- focal shells, 256. External particle in the plane of the equator, 257. Maclaurin's results, 259. His investigations under-estimated, 260. At- traction of an oblatum, 261. Application to rotating fluid, 262. Variable density, 264. Attraction of oblatum of varying density, 266. Applica- tion of Newton's Hydrostatical principle, 267. Objections, 269. Polar and equatorial columns, 270. Jupiter, 272. Essay on the Tides, 275. Maclaurin's death, 276. CHAPTER X. THOMAS SIMPSON . . . .176 Mathematical Dissertations, 277. Reference to Maclaurin, 278. Attraction of an oblatum, 279. Value of a definite integral, 280. Values of two definite ^ integrals, 281. Relation between excentricity and the angular velocity, 283. Limit of angular velocity, 284. Two solutions, 285. Conservation of Areas, 286. Oblatum not homogeneous, 289. Attraction of Spheroids, 290. Length of a Degree, 292. Fluxions, 293. Simpson's eminence as a mathematician, 294. CHAPTER XI. CLAIRAUT 189 Figure de la Terre, 295. Fluid equilibrium, 296. Cartesians and New- tonians, 297. Principle of Canals and Principle of Level Surfaces, 298. Points of interest in the Introduction, 299. Clairaut's first part, 300. Comparison between Clairaut and others, 301. Principle of Canals, 302. T. M. A. c XX11 CONTENTS. PAGE General reasoning, 303. Cases where equilibrium is impossible, 304. Principles of Newton and Huygens, 305. Rotating fluid, 306. Complete differential, 307. Principle of Level Surfaces, 308. Examples of fluid equilibrium, 309. Bouguer's problem, 310. Polar Coordinates, 312. Space of three dimensions, 313. Capillary attraction, 314. Hetero- geneous fluid, 315. Law of attraction, 316. Clairaut's second part, 317. Homogeneous Figure of the Earth, 318. Jupiter, 319. Attraction of a circular lamina, 320. Attraction of an oblatum, 321. Ellipticity, 322. Fluid about a solid nucleus, 323. Particular cases, 324. Criticisms on Newton, 325. Oblongum may be a possible form, 326. Case in which the thickness of fluid is small, 327. Objection by D'Altmbert and Cousin, 328. The elliptic! ty is in general less when the fluid is heterogene- ous than when it is homogeneous, 329. Variation of gravity, 330. Attrac- tion of a circular lamina, 331. Of an oval lamina, 332. Proposition as to the attraction of an ellipsoid of revolution, 333. Particle on the pro- longation of the axis of revolution, 334. Analytical verification of a result obtained by Clairaut, 335. Gravity at the surface of the Earth: Clairaut's Theorem, 336. Inaccurate statements which have been made, 337. Applications of Clairaut's theorem, 338. Strata of varying density, 339. Attraction of a shell, 340. Primary Equation, 341. Derived Equa- tion, 343. Another form of the derived equation, 344. Example of a particular law of density, 345. Clairaut's derived equation, 346. Case in which the mass consists of fluid of two densities, 347. Limits for the ellipticity of a planet, 348. Comparison of theory with observation, 349. Laplace's opinion, 350. CHAPTER XII. ARC OF THE MERIDIAN MEASURED IN PERU 231 The Peruvian expedition started before the Lapland, 35 1 . Literature of the subject, 352. Course of the operations, 353. Difficulties encountered, 354. Base of verification, 355. The astronomical part, 356. Discordance of the observations, 357. Final result, 358. The Spanish operations, 359. Return to Paris, 360. Miscellaneous points, 361. The Spanish ac- count, 362. Bouguer's Figure de la Terre, 363. Summary, 364. CHAPTER XIII. D'ALEMBERT . Treatise on Fluids, 365. Historical sketch, 366. Comparison between theory and actual measurement, 367. Second edition, 368. Matters of in- terest, 369. Treatise on the Winds,' 370. Two sentences quoted, 371. Companion to Huygens's problem, 372. Attraction of a homogeneous oblatum at its surface, 373. Spherical nucleus, 374. Results which D'Alembert considers strange, 375. Oblate nucleus, 376. Particular case of a formula given by Clairaut, 377. Criticism on Clairaut, 378. CONTENTS. XX 111 PAGE Incorrect statement, 379. Attraction at the surface of an ellipsoid, 380. Error, 381. Contradiction and error, 382. Ellipsoidal nucleus, 383. Precession of the Equinoxes, 384. Important numerical relation, 385. Combination of D'Alembert's result with Clairaut's, 386. Example in the Integral Calculus, 387. Hypothesis as to the structure of the Earth, 390. Inequality, 391. Treatise on the Resistance of Fluids, 392. His hydro- statical principle, 394. Failure in attempt at generalization, 397. Sur- faces of equal density not necessarily level surfaces, 400. Failure in attempt to generalize Clairaut's theory, 404. Researches on the System of the World, 409. Criticism on Euler, 411. Difference between theory and observation, 416. Results of integration, 423. General formula for the attraction of a spheroid, 424. Spherical nucleus, 426. Oblate nucleus, 429. Criticism on Clairaut, 431. Attraction of spherical shell, 434. Results of integration, 436. Attraction of a spheroid of revolution, 437. Three general equations, 444. General estimate of his researches, 452. - CHAPTER XI Y. BOSCOVICH AND STAY . . .305 The work by Boscovich and Le Maire, 454. Eoscovich's dissertations, 457. Follows Maclaurin's solution, 461. Criticises Hermann, 466. Criticises D. Bernoulli, 469. Pendulums, 472. Suggestion, 476. Measures of a degree of the meridian, 481. Criticises Euler, 483. Abstract of the treatise, 489. Stay's poem, 490. Dugald Stewart's opinion, 491. Specimens of Stay's verses, 493. Supplementary dissertations, 494. Newton's error, 499. D. Bernoulli's opinion of Newton, 501. Measures of arcs, 508. Criticises Maupertuis, 510. Method for discordant obser- vations, 511. French translation, 513. CHAPTER XV. MISCELLANEOUS INVESTIGATIONS BE- TWEEN THE YEARS 1741 AND 1760 . . .335 Pendulums, 515. Murdoch, 516. Bremond's translation, 521. Addition by Murdoch, 522. Mairan, 526. Cassini de Thury, 527. Clairaut and Buffon, 528. E. Zanottus, 529. La Condamine, 530. Silvabelle, 531. Frist and Short, 532. Defence of Newton, 534. Clairaut's reply to Frisi, 535. Bouguer, 538. La Caille's voyage, 53$. La Caille's arc, 541. Maclear's arc, 542. La Lande, 543. Euler, 545. La Caille's reply to Euler, 546. Hollmannus, 547. Euler, 548. La Caille, 549. La Condamine, 550. Pieard's base, 551. Walmesley, 552. La Caille, 553. D'Arcy, 554. Clairaut's Prize Essay, 555. Edition of the Principia, by Madame du Chastellet, 558. Lagrange on D'Alembert's paradox, 561. CHAPTER XVI. D'ALEMBERT 365 Articles in the Encyclopedic, 564. Reply to Boscovich, 567. Paradox, 569. Corrects former errors, 571. Fluid equilibrium, 574. Attempts to solve c2 XXIV CONTENTS. PAGE Legendre's problem, 575. Failure, 576. Sixth volume of the Opuscules Mathe'matiques, 579. Two solutions of the problem of rotating fluid, 580. The case of a very small angular velocity, 584. Only two solutions, 586. Replies to the translator of Boscovich, 590. Attraction of mountains, 592. Discussion of a problem, 596. An oblongum not an admissible form, 601. Two analytical matters, 6n. Generalisation of a former problem, 613. Correction of a recent error, 618. Attraction of an ellipsoid, 625. Ex- tension of former problem, 629. Error, 630. Criticises Clairaut, 634. Considers Maclaurin's theorem, 636. Atmosphere, 637. Equation to the surface, 639. Letters to Lagrange, 643. Demonstrates Maclaurin's theorem, 645. Rejects an important formula, 651. Investigates a theo- rem given by Laplace, 652. Fluid equilibrium, 654. Unsound demon- stration, 657. Summary of his contributions, 658. CHAPTER XVII. FRISI 424 De Gravitate, 660. Measurements hitherto made, 661. Extends a result of Newton's, 662. Criticism on Newton, 663. Newton's error, 664. Cos- mographia, 668. Suggestion as to La Caille's arc, 671. Mistake corrected, 673. Newton's error, 676. Stability of equilibrium, 678. Fontana, 679. Opera, 680. Silvabelle's problem, 682. CHAPTER XVIII. MISCELLANEOUS INVESTIGATIONS BETWEEN THE YEARS 1761 AND 1780 . . . 439 Frisi, 686. Krafft, 687. Error, 689. Sum of a series, 694. Osterwald, 696. Michell, 697. Cauterzanus, 698. E. Zanottus, 699. J. A. Euler, 700. Lambert, 701. Liesganig, 702. Mason and Dixon, Maskelyne, 703. Liesganig, 704. Cavendish, 705. La Condamine, 706. Lagrange, 707. Transformation of variables, 710. Ellipsoid, 712. Beccaria, 717. Cassini de Thury, 718. Lagrange, 720. Demonstrates Maclaurin's theorem, 721. Cassini de Thury, 723. Schehallien experiment, 724. Statement by Newton, 725. Dionis du Sejour, 728. Whitehurst, 729. Hutton's calculations for Schehallien, 730. Playfair's survey of the mountain, 731. Hutton, 732. Density of the Earth, 733. Cousin, 734. Euler, 735. Hutton, 736. Titles of works, 738. Additional remarks, 740. CONTENTS. XXV VOLUME II. PAGE CHAPTER XIX. LAPLACE'S FIRST THREE MEMOIRS . 1 Five divisions in Laplace's writings, 741. First Memoir, 742. Legendre's problem, 744. Second memoir, 751. Law of variation of gravity, 753. Laplace's equation, 755. Third memoir, 764. Mistake, 769. Laplace's equation, 771. Order of the writings of Laplace and Legendre, 778. CHAPTER XX. LEGENDRE > S FIRST MEMOIR . . 20 Treatment of Maclaurin's theorem, 781. Extension of it, 782. Laplace's coeffi- cients, 783. Heine's work, 784. General expression for a coefficient, 786. Potential, 789. Green and Gauss, 790. Legendre's theorem, 791. Cri- ticism on the demonstration, 792. Extension of Maclauriu's theorem, 793,. Character of the memoir, 794. CHAPTER XXI. LAPLACE'S TREATISE. ... 29 Its scarcity, 796. Publication, 797. Number of sections, 800. Equation to an ellipsoid, 801. Potential, 802. Polar coordinates, 803. Laplace's theorem, 804. Attraction of an ellipsoid, 805. History of the problem, 806. Rotating fluid mass, 807. Treatment of D'Alernbert's problem, 808. Case of the Moon, 809. Case of the Earth, 810. Two and only two solutions, 811. Conservation of areas, 813. Laplace's equation, 814. General problem of the form of fluid, 815. Law of gravity at the surface of a fluid mass, 816. Spherical shell attracting external particle, 817. CHAPTER XXII. LEGENDRE'S SECOND MEMOIR . . 43 Object of the memoir to solve Legendre's problem, 820. Conditions of the demonstration, 821. Seven theorems as to Laplace's coefficients, 822. Equation to be solved, 831. Mode of treatment, 837. Remarks on the demonstration, 842. Legendre's own opinion respecting it, 844. Quota- tion from Laplace, 845. Quotation from Ivory, 846. Quotation from Jacobi, 847. CHAPTER XXIII. LAPLACE'S FOURTH, FIFTH, AND SIXTH MEMOIRS 55 Fourth memoir, 848. Laplace's theorem, 850. Partial Differential Equation for the Potential, 851. Laplace's Equation, 852. Error, 854. Property of Laplace's functions, 857. Fifth memoir, 859. Degrees and Pendulums, 860. Numerical example, 862. Oversight, 863. Sixth memoir, 864. Saturn's ring, 865. Partial Differential Equation for the Potential, 866. Division of Haturn's ring, 867. Form of the ring, 868. Plana's criticisms on Laplace, 871. Instability of a riug, 872. XXVI CONTEXTS. PAfi E CHAPTER XXIV. LEGENDRE'S THIRD MEMOIR . . 74 Object of the memoir to demonstrate Laplace's Theorem, 875. Transformation of multiple integrals, 877. Case in which the attracted particle is in a principal plane of the ellipsoid, 880. General problem, 88 1. Point of the Integral Calculus, 882. Legendre's theorem, 883. Limitation in Legeudre's process, 886. Opinions of the method, 887. CHAPTER XXV. LEGENDRE'S FOURTH MEMOIR . . 87 Object of the memoir, 892. Error, 896. Laplace's coefficients, 897. First hypothesis, 902. Result of first approximation, 909. Second approxima- tion, 912. Ellipticity, 917. Length of degree, 918. General theorem, 920. Force of gravity, 921. Clairaut's theorem, 923. Second hypothesis, 925. Clairaut's equation generalised, 929. Vanishing of terms, 933. Ex- amples of laws of density, 939. First example ; density constant, 940. Second example, 941. Third example, 94-2. Second approximation, 943. Third hypothesis, 944. Numerical values, 945. Case in which the figure is npt assumed to be one of revolution, 948. Laplace's functions, 949. Correction of an oversight in Laplace, 953. The form of a planet must be an oblatum, 955. General estimate, 957. CHAPTER XXVI. LAPLACE'S SEVENTH MEMOIR . . 131 Mode of treating measured lengths of degrees, 960. Another method due to Boscovich, 962. Lengths of seconds pendulum, 964. Two methods of treatment, 966. Clairaut's Theorem, 967. CHAPTER XXVII. MISCELLANEOUS INVESTIGATIONS BETWEEN THE YEARS 1781 AND 1800 . . .138 Euler, 970. Krafft, 971. Error of B. S* Pierre, 972. Cousin's Elementary Treatise, 973. Error, 976. Approximate formulae, 978. Attempts to generalise Clairaut's theory, 980. Error, 982. Roy, 984. La Lande, 985. Roy, 986. Embarrassed by an error of Bouguer's, 987, Williams, 988. La Lande on FerneFs measure, 989. Legendre's Theorem in Spherical Trigonometry, 990. Monge, 991. Borda and others, 992. Coulomb's theorem, 993. Lagrange's Me'canique Analytique, 994. Waring, 995. Triesnecker, 996. Pictet, 997. Waring, 998. Dalby, 999. Cassini IV. and others, 1000. Cagnoli and Baily, 1001. Topping, 1002. LaLande's Astronomic, 1003. Lagrange, 1004. Partially investigates Laplace's theorem, 1008. Rumovsky, 1012. Prony, 1014. Cavendish, 1015. Result as to the mean density of the Earth, 1018. Trtmbley, 1019. Algebraical identity, 1024. Error, 1027. Legendre's coefficients, 1028. Laplace's equation, 1030. Fontana, 1034. Van-Swinden, 1035. Survey of England and Wales, 1036. General Roy's rule, 1037. C< INTENTS. XXV11 PAGE CHAPTER XXVIII. LAPLACE, ME*CANIQUE CELESTE, FIRST AND SECOND VOLUMES . , . .176 First volume, 1041. Potential, 1042. Case of a spherical shell, 1043. Objection, 1044. Spherical shell and external particle, 1045. Laplace's enunciation extended, 1046. Spherical shell and internal particle, 1047. Cylinder, 1048. Cylindrical shell, 1050. Second volume, 1052. First Chapter, Spheroid, 1053. Attraction of an ellipsoid, 1054. Laplace's theorem, 1060. Professor Cay ley's paper, 1 06 1. Second Chapter, Laplace's functions, 1064. Various names for them, 1066. Any function may be expanded in a series of them, 1067. Form of the expansion, 1069. Legendre's theorem extended, 1076. Third Chapter, Oblatum, 1078. His results, 1079. Sources of them, 1080. Oblongum inadmissible, 1082. One oblatum for given moment of rotation, 1085. Fourth Chapter, 1088. Bouguer's hypothesis rejected, 1094. Approximation extended, 1098. Fifth Chapter, iioo. Misprints, 1102. Fifth Chapter, 1103. Fil ' sfc method of calculation, 1104. Second method, 1105. Jupiter, 1109. Lapland arc, mi. Arc perpendicular to the meridian, 1112. Sixth Chapter, Saturn's ring, 1116. Attraction of an elliptic cylinder, 1119. The resultant constant at the surface, 1123. Seventh Chapter, Atmo- sphere, 1126. Summary of results, 1127. CHAPTER XXIX. LAPLACE'S THEOREM . . .216 History of the Theorem, 1 129. Biot's investigation, 1 130. General theorem, 1 133. Particular case, 1136. Biot's historical statement questioned, 1138. Ivory, 1140. His enunciation, 1143. Opinions of his merit, 1146. Plana, 1147. Biot's appendix, 1148. Legendre, 1149. History of the theorem, 1150. Expressions for the attraction of an ellipsoid, 1152. Approximation for a nearly spherical body, 1155. Elliptic integrals, 1156. Algebraical relation, 11 = 7. Another relation, 1158. Poisson extends Ivory's theorem, 1160. This does not apply to Laplace's theorem, 1161. Gauss, 1162. His fifth theorem, 1169. His fourth theorem, 1170. His third theorem, 1171. His sixth theorem, 117-2. Application to the ellip- soid, 1173. Gauss's reference to Ivory, 1175. Rodrigues, 1176. Rela- tive Potential, 1179. Symmetrical expression for the Potential of an ellipsoid, 1184. Formula for Legendre's coefficient, 1187. General pro- perty of the coefficients, 1189. Summary of results, 1194. CHAPTER XXX. LAPLACE'S EQUATION . . .253 Particular form, 1196. Lagrange, 1197. The difficulty which he explains, 1200. Ivory's objections, 1203. History of the equation, 1204. The point of diffi- culty, 1207. Expansion in a series of Laplace's functions, 1213. General form of the equation, 1214. Ivory's objections, 1215. Ivory's notice of Lagrange's memoir, 1216. Second memoir by Ivory, 1219. Laplace's later investigation, 1220. Poisson, 1223. Ivory returns to the sub- XXV111 CONTENTS. PAGE ject, 1224. Laplace's opinion of Ivory, 1226. Airy, 1227. His treat- ment of the equation, 1228. Expansion in a series of Laplace's func- tions, 1230. Me Cullagh, 1233. Plana, 1235. CHAPTER XXXI. PARTIAL DIFFERENTIAL EQUATION FOR V 274 Laplace's equation, 1236. Poisson's correction, 1237. Applications, 1239. Rodrigues, 1240. Poisson's later method, 1241. The three cases, 1244. Objection, 1245. Ostrogradsky, 1247. Extends the unsatisfactory case given by Poisson, 1248. Sturm, 1251. Bowditch, 1252. Gauss's cri- ticism, 1253. CHAPTER XXXII. LAPLACE'S SECOND METHOD OF TREATING LEGENDRE'S PROBLEM .... 284 History of the problem, 1254. Liouville, 1255. Laplace's process, 1256. Limits of the integration, 1261. Liouville's objection to Laplace's process^ 1263. Poisson's process, 1265. Remarks on it, 1267. Laplace's supplementary investigation, 1270. Wantzel, 1272. Transformation of a double integral, 1273. CHAPTER XXXIII. LAPLACE'S MEMOIRS . . . 805 Saturn's ring, 1275. Rotation of the Earth, 1276. Figure of the Earth, 1277. Rotation of the Earth, 1278. Theorem as to a principal axis, 1281. Law of gravity, 1283. Figure of the Earth, 1284. Cooling of the Earth, 1287. Increase of temperature with increase of depth, 1290. Mean density of the Earth, 1291. Extracts on Hutton, Cavendish and Newton, 1292. CHAPTER XXXIV. FIFTH VOLUME OF THE MECANIQUE CELESTE 315 Eleventh Book, 1294. First Chapter, Historical sketch, 1295. Elephant preserved in the ice, 1296. Second Chapter, Results obtained by Ana- lysis, 1301. Value of gravity, 1305. Expression for the depth of the sea, 1310. Value of gravity, 1313. Approximate values of Legendre's functions, 1314. Variations of the lengths of degrees and of the value of gravity, 1316. Expression for gravity at the surface of a supposed atmo- sphere, 1318. Comparison of the analysis with observations, 1320. Lunar Theory, 1322. Measures of degrees, 1323. Precession and Nutation, 1324. Hypothetical law connecting the pressure and the density, 1325. Legendre's law of density, 1326. Numerical results, 1328. Young and D. Bernoulli, 1330. Attraction of a mountain, 1332. Third Chapter, 1336. Theorem as to the three principal axes, 1338. Transformation of angular coordinates, 1340. Fourth Chapter, 1345. Fourier and Poisson, 1346. Analytical results, 1350. Correction, 1351. General summary, 1354. CONTENTS. PAGB CHAPTER XXXV. POISSON 349 List of his writings, 1356. Memoir on electricity, 1357. Attraction of sphe- roids, 1358. Laplace's functions, 1360. Value of the potential, 1365. Accurate distinction of cases, 1364. Partial differential equation for the potential, 1365. Spherokls which differ little from spheres, 1367. A point hitherto neglected now examined, 1368. Investigation carried to a certain order, 1369. May be extended, 1371. Two forms coincide at the surface, 1372. A transformation, 1373. Laplace's equation extended, 1374. Application to rotating fluid, 1375. Novelties in the process, 1376. Ap- proximation to the second order, 1378. Coulomb's theorem, 1380. Hetero- geneous fluid, 1381. Partial differential equation for the potential, 1382. Ivory's criticism, 1384. Addition to the memoir, 1385. Double inte- gral, 1386. Argument against Ivory, 1388. Convergence of series, 1389. Poisson's Mechanics, 1390. Attraction of ellipsoids, 1391. Remark- able result, 1394. New forms for the component attractions, 1395. Elliptic integrals, 1398. Note on a result given by Jacobi, 1401. Con- troversy between Poisson and Poinsot, 1404. Poisson's remarks on Poinsot's report, 1405. Poin sot's reply, 1408. Poisson's Addition, 1410. Poinsot' s reply, 1411. Remarks on the controversy, 1412. Note on the general formula of attractions, 1413. Liouville's process of investiga- tion, 1414. General summary, 1415. CHAPTER XXXVI. IVORY 391 List of his writings, 1416. Attraction of an extensive class of spheroids, 1417. Expansion in a series, 1420. Equilibrium of a fluid, 142 1. New principle as- sumed, 1422. Theorem on the Potential, 1424. Inconclusive reasoning, 1425. True proposition, 1426. Proposition which is not necessarily true, 1427. Error, 1428. Converse of a known result, 1429. Supposed solution of Legendre's problem, 1430. Good treatment of a standard equation, 1432. Unsupported assertion, 1433. Unintelligible reason, 1434. Article on Attraction, 1435. Figure of the Earth, 1436. Fluid attracted to a fixt centre, 1441. Proposed second approximation, 1442. Opinion on the Theory of the Figure of the Earth, 1444. Criticisms on Professor Airy and Poisson, 1445. Pronounces a certain theorem inaccurate, 1448. Demonstrates the theorem, 1449. Criticises a remark made by Biot, 1453. Erroneous statements, 1456. Ivory's assumed principle, 1459. Discusses erroneously Jacobi's theorem, 1460. Remarks on Poisson, 1461. General summary, 1464. CHAPTER XXXVII. PLANA 413 List of his writings, 1465. Commentary on Lagrange, 1466. Problems in Attraction, 1468. Erroneous result, 1472. Solution of a problem, 1474. Neglect of the principle of dimensions, 1476. Saturn's ring, 14/9. Solution of a problem, 1480. Two definite integrals, 1484. Law of XXX CONTENTS. FAGB density and pressure, 1486. Formula given by Gauss, 1489. Law of density, 1491. Attraction of ellipsoid, 1492. Opinion of Legendre's pro- cess, 1495. Criticises Legendre and Pontecoulant, 1498. Remark on Rodri- gues, 1502. Unsupported statement, 1503. Remark on Legendre, 1505. Refers to the example of Euler, 1507. Poisson and Legendre, 1508. The potential of an ellipsoid, 1509. Appendix to the memoir on the attraction of an ellipsoid, 1513. Notes on Newton, 1515. Opinion on Newton's method, 1519. Density of the superficial stratum, 1520. Diffi- culties, 1524. Density of a mountain, 1527. Force of gravity, 1528. Hypothesis of Huygens, 1529. Criticises Laplace, 1530. Attributes an error to Newton, 1534. Criticises Calandrini, 1535. Approximate so- lution, 1538. Oblongum is not an admissible figure, 1539. Jacobi's Theorem, 1540. Potential of an ellipsoid, 1543. Criticises Poisson, 1544. Hypothesis of uniformly increasing density, 1546. Formula of the Integral .Calculus, 1550. Gravity at the surface of the sea, 1551. Laplace's equation, 1553. Remark on D'Alembert, 1556. Tides, 1558. Remark on Newton, 1559. General summary, 1560. CHAPTER XXXVIII. MISCELLANEOUS INVESTIGATIONS BETWEEN THE YEARS 1801 AND 1825 . . . 453 Clay, 1562. Benzenberg, 1563. Burckhardt's translation of Laplace, 1564. Other works of the same nature, 1565. Playfair, 1566. Length of arc of the meridian, 1568. The Survey of England, 1572. Svanberg, 1575. Von Zach, 1576. Base du Systeme Me"trique, 1577. Dr Young, 1578. De Zach, 1579. Lagrange, 1580. Example of Fluid equilibrium, 1581. Error, 1582. Playfair, 1583. Silvabelle's problem, 1584. Error, 1587. General result, 1589. Knight, 1593. Attraction of a right-angled triangle, 1595. Extension of Silvabelle's problem, 1599. Modification of the problem, 1601. Formula of the Integral Calculus, 1603. Solution of an example, 1604. De Zach on the Attraction of mountains, 1605. Quotations, 1607. Tadino, 1608. Cauchy, 1609. Dissertation on pendu- lums, 1610. Luckcock, 1611. Adrain, 1612. Lambton's Indian arc, 1614. Nobili, 1615. Dr Young, 1616. The mean density of the Earth, 1617. Dr Young's Rule, 1618. Ellipticity calculated, 1619. Erroneous solution of a problem, 1621. General remarks, 1623. Wronski, 1624. Airy, 1625. Saturn's ring, 1627. Different numerical calculations, 1628. Form of Saturn, 1630. Beccaria's arc, 1631. Bowditch, 1632. CHKONOLOGICAL LIST OF AUTHORS. The figures refer to the Articles of the Volumes. ART. 1637 NORWOOD 68,71 1669 PlCARD 7O, I O2 ElCHER 69 VARIN 69 DBS HAYES 69 Du GLOS 69 1686 HALLET 72 1687 NEWTON i 1690 HUYGENS 47 1691 ElSENSCHMIDT 76 1698 KEILL 73 1699 KEILL 79 1700 COUPLET 80 1701 DES HAYES 81 1701 D. CASSINI 81 1702 J. CASSINI 82 1 702 D. GREGORY 84 1703 DELAHIRE 83 1708 KEILL 85 1708 FEUILLEE 90 1711 FREIND 91 1713 NEWTON i 1713 J. CASSTNI 92 1716 HERMANN 93 1718 J. CASSINI 99 1720 J. CASSINI 100 1720 MAIRAN 109 1725 DESAGULIERS 200 1726 NEWTON i 1728 POLENI 209 1729 DE LA CllOYERE 2IO 1732 MAUPBRTUIS 117, 122, 128 1732 J. CASSINI -2U ART. 1733 MAUPERTUIS 131 1733 GODIN 213 1733 LA CONDAMINE 2(4 ^733 J- CASSINI 215 1733 CLAIRAUT 160 1734 MAUPERTUIS . 132 1734 BRADLEY 217 1734 MANFREDI 218 1734 BOUGUER 219 1734 J.CASSINI 220 1734 JOHN BERNOULLI 221 1735 MAUPERTUIS 140 1735 STIRLING 151 1735 CLAIRAUT 161 1735 J. CASSINI 222, 223 1735 CASSINI DE THURY 224 1735 MAIRAN 225 1735 GODIN 225 1735 BOUGUER 225 1735 LA CONDAMINE 225 1736 MAUPERTUIS 141 1736 CLAIRAUT 162 1736 CASSINI DE THURY 226 1736 BOUGUER 227 1737 MAUPERTUIS 142 1737 CLAIRAUT 163 1737 DELISLE 228 1738 MAUPERTUIS 146, 150, 181 1738 CLAIRAUT 167 1738 CELSIUS i8r, 198 1738 EULER 229 1738 D. BERNOULLI 230 1739 CLAIRAUT 177 XXX11 CHRONOLOGICAL LIST OF AUTHORS. ABT 1739 CASSINI DE THUBT 231 1740 D. BEBNOULLI -. 1740 EULEB 234 1740 MAUPEBTUIS 235 1740 CASSINI DE THUBY 236 1740 WINSHEIM 240 1740 MACLAUBIN 275 1741 MUBDOCH : 1742 ZELLEB 181 1742 LE SEUB 239 1742 JACQUIEB i 1742 CALANDBINUS 239 1742 MACLAUBIN 241 1742 BBEMOND 521 1742 MAIBAN 526 1743 SIMPSON 277 1743 CLAIBAUT 295 1744 OUTHIEB 182 1744 BOUGUEB 352 1744 D'ALEMBEBT 365 1744 CASSINI DE THUBY 527 1745 LA CONDAMINE 352 1745 BUFFON 528 1745 CLAIBAUT 528 1746 BOUGUEB 352 1746 LA CONDAMINE 352 1746 ZANOTTUS, E 529 1747 D'ALEMBEET 370 1747 LA CONDAMINE 530 1748 JUAN 352 1748 ULLOA 352 1748 CASSINI DE THUBT 530 1749 BOUGUEB 352 1749 D'ALEMBEBT 384 1749 EULEB 411 1750 SIMPSON 293 1750 SlLVABELLE 531 1751 LA CONDAMINE 352 1751 BOUGUEB 538 1751 LA CAILLE 539 1752 BOUGUEB 352 1752 LA CONDAMINE.: 352 1752 D'ALEMBEBT 392 1752 LA CAILLE 539 1752 LA LANDE 543 1753 FBISI 532 1753 SHOET 535 ABT. 1753 CLAIBAUT 535 1753 LA LANDE 544 1753 EULEB 545 1754 BOUGUEB 352 1754 LA CONDAMINE 352 1754 D'ALEMBEBT 410 1754 LA CAILLE 546 1754 HOLLMANNUS 547 1754 BOUGUEB and others 551 1755 BOSCOVICH 454 1755 EULEB 548 1755 LA CAILLE 549 1756 D'ALEMBEBT 415, 564 1756 LA CONDAMINE 550 1757 BOSCOVICH 489 1757 D'ALEMBEBT 565 1758 WALMESLET 552 1758 LA CAILLE 553 1758 D'ABCY 554 1759 CLAIBAUT 555 1759 CHASTELLET 558 1759 LAGBANGE 561 1760 BOSCOVICH 490 1760 STAY 490 1760 LAGBANGE 562 I/6l D'ALEMBEBT 566 1761 FBISI 686 1763 LA CAILLE 540 1764 KBAFFT 687 1764 OSTERWALD 696 1/66 MlCHELL 697 1767 CANTEBZANUS 698 1767 E. ZANOTTUS 699 1767 LAMBEBT 701 1768 D'ALEMBEBT 570 1768 FBISI 660 1768 J. A. EULEB 700 1768 LIESGANIG 702 1768 MASON 703 1768 DIXON 703 1768 MASKELYNE 703 1770 D'ALEMBEBT 368 I77O LlESGANIG 704 1772 CAVENDISH 705 1772 LA CONDAMINE 706 772 LAPLACE 751 773 D'ALEMBEBT 579 CHRONOLOGIC AT, LIST OF AUTHORS. XXX111 1773 i/73 1774 '774 1775 1775 1775 1775 1775 1776 1778 1778 1778 1778 1779 1780 1780 1782 1783 1784 1784 1784 1784 1784 1785 1785 1785 1785 1786 1787 1787 1787 1787 1787 1787 1788 1788 1788 1788 1788 1788 1788 1788 1789 1789 1789 1791 ART. LAGRANGE 707 LAPLACE 742 D'ALEMBERT 643 BECCARIA 717 FRISI 668 LAGHANGE 720 CASSINI DE THURY 723 MASKELYNE 724 LAPLACE 764 CASSINI DE THURY 718 LA CONDAMINE 352 WHITEHURST 729 BUTTON 730 COUSIN 734 EULER 735 D'ALEMBERT 644, 654 HUTTUN 736 LAPLACE 848 LAPLACE 859 EULER 970 KRAFFT 97! ST PIERRE 972 LAPLACE 795 LEGENDRE 819 FRISI ...-. . 680 EOY LA LANDE LEGENDRE WILLIAMS COUSIN ROY LA LANDE LEGENDRE MONGE LAPLACE 864 LEGENDRE 874 WILLIAMS 988 BORDA and others 992 CASSINI IV 992 BRISSON 002 LEGENDRE COULOMB LAGR ANGE LEGENDRE LAPLACE WARING... 984 985 779 988 973 986 989 990 991 992 993 994 891 958 995 TBIESNECKER 996 r79f PlCTET ........................ .. 1791 WARING 1791 DALBY 1791 CASSINI IV ................... 1791 MECHAIN ........................ 1791 LEGENDRE ..................... 1792 CAGNOLI ........................ 1792 TOPPING ........................ 1792 LA LANDE ..................... 1792 LAGRANGE ..................... 1796 RUMOVSKY .................. '... 1797 PRONY ........................... 1798 CAVENDISH ..................... 1799 TREMBLEY ..................... 1799 FONTANA ........................ 1799 VANSWINDEN .................. 1799 LAPLACE ........................ 1802 CLAY ........................... 1802 BURCKHARDT .................. 1804 BENZENBERG .................. 1805 PLAYFAIR ..................... 1805 SVANBERG ..................... 1806 BIOT .............................. 1806 A. VON ZACH .................. 1806 DELAMBRE ..................... 1807 LAPLACE ........................ 1808 YOUNG ........................... 1809 IVORY ........................... 1809 LAGRANGE ..................... 1810 LEGENDRE ..................... 1810 DE ZACH ....................... 1811 DE ZACH ........................ 1811 PLAYFAIR ..................... 1811 PLANA ........................... l8ll POISSON ........................ 1811 LAGRANGE ..................... 1812 BIOT .............................. 1812 POISSON ........................ 1812 IVORY ............ 1203, 1219, 1812 PLANA ........................... 1812 PLAYFAIR ........ ............. 1812 KNIGHT ........................ 1813 GAUSS ........................... 1813 POISSON ........................ 1814 DE ZAOH ........................ 1814 TADINO ........................... 1815 CAUGHT ........................ ART. 997 998 999 1000 1000 1000 1002 1003 1004 1012 1014 1015 1019 1034 1035 1040 I5 6 2 1564 1563 1575 II30 1576 ^577 1275 1I4 o 1197 1149 717 73! 1357 1580 II4 8 1417 1605 XXXIV CHRONOLOGICAL LIST OF AUTHORS. ART. 1815 T ENGSTROM and BONSDORFF 1610 1816 RODRIGUES 1176, 1240 1817 LAPLACE 1220, 1286 1817 LUCKCOCK 1611 1818 PLANA 867 1818 LAPLACE 1276, 1277, 1286 1818 ADRAIN 1612, 1613 1818 LAMBTON 1614 1818 NOBILI 1615 1819 BAILT 1001 1819 L'APLACE 1284 1819 YOUNG 1616 1820 LAPLACE 1287, 1288 1820 PLANA 1468 1820 YOUNG 1619 1820 WBONSKI 1624 1821 BUTTON 733 1821 LAPLACE 1278, 1283 1821 PLANA 1486 1821 BIOT and ARAGO 1577 1822 IVORY 1224, 1419 1822 LAPLACE 1285 1823 POISSON 1223, 1241 1823 LAPLACE 1289, 1291 1824 CARLINI 733 1824 IVORY 1421, 1435, 1436 1824 POISSON 1422 1825 LAPLACE 1293 1825 IVORY 1437, 1438, 1439 1825 BOWDITCH 1632 1826 IVOBY 1439 to 1442 1826 YOUNG 1621 1827 AIRY 1227,1625 1827 POISSON 1384 1827 BIOT 1577 1827 IVORY 1443 to 1445 1828 IVORY 1446 to 1448 1829 PONTECOULANT 1231 1829 POISSON 1246, 1358 1829 IVORY 1449101451 1830 SCHMIDT 733 1830 STURM 1251 ART. 1830 IVORY 1452 to 1454 1831 OSTROGRADSKY 1247 1831 POISSON 1385 1831 IVORY 1455 1832 BOWDITCH 1232 1833 POISSON !39 1835 POISSON 1391 1837 MACCULLAGH 1233 1837 LIOUVILLE 1255 1837 POISSON 1265, 1400 1838 REICH 733 1838 POISSON 1404, 1413 1838 POINSOT 1404 1838 LIOUVILLE 1414 1838 IVORY 1460, 1461 1839 WANTZEL 1272 1839 IVORY 1460, 1462 1840 MENABREA 733 1840 GIULIO 733 1840 GAUSS 1253 1840 PLANA 1492, 1509 1841 R.L.ELLIS 1422 1843 BAILY 733 1843 PLANA 1513 1843 BIOT 1577 1845 PLANA 1357 1845 BENZENBERG 1563 1847 HEARN 733 1849 CAYLEY 890 1851 PLANA 1515 1852 REICH 733 1852 PLANA 1520, 1529 1853 PLANA 1533, 1546 1854 PLANA 1551 1855 YOUNG. 1622 1856 AIRY 733 1856 HAUGHTON 733 1857 CAYLEY 1060,1163 1858 CAYLEY 1178 1861 HEINE 784 1866 MACLEAR 542 1869 SCHELL 733 XXXV THE following Table gives the DATES OF BIRTH AND DEATH of the principal writers on ATTRACTION and the FIGURE OF THE EARTH : BIOT 1777 1862 BOSCOVICH 1711 1787 BOUGUEB 1698 1758 BOWDITCH 1773 1838 CASSINI, J. D 1625 1712 CASSINI, J 1677 1756 CASSINI DE THURT 1714 1784 CASSINI IV 1748 1845 CAVENDISH 1731 1810 CLAIRAUT 1713 1765 COULOMB 1736 1806 COUSIN 1739 1800 D'ALEMBERT 1717 1783 DELAMBBE 1749 1822 EULER 1707 1783 FRISI 1728 1784 GAUSS , 1777 1855 HUYGENS 1629 1695 IVORY 1765 1842 LACAILLE 1713 1762 LA CONDAMINE 1701 1774 LAGRANGE 1736 1813 LA LANDE 1732 1807 LAPLACE 1749 ^27 LEGENDRE 1752 1833 MACLAURIN 1698 1746 MAIRAN 1678 1771 MASKELYNE 1732 i8n MAUPERTUIS 1698 1759 NEWTON 1642 1727 PLANA 1781 1864 PLAYFAIB 1748 1819 POISSON 1781 1840 SIMPSON 1710 1761 XXXVI THE following Table gives references to the principal numerical discussions of the Figure and Dimensions of the Earth in chronological order. 1755 BOSCOVICH. De Litteraria Expeditione. 1760 BOSCOVICH. Stay's Philosophies Recentioris. 1768 FBISI. De Gravitate. 1770 French translation of Boscovich's work. 1783 LAPLACE'S fifth Memoir. 1789 LAPLACE'S seventh Memoir. 1799 LAPLACE. Mecanique Celeste, Vol. II. 1826 AIRY. Philosophical Transactions. 1828 IVORY. Philosophical Magazine. 1830 AIBY. Encyclopaedia Metropolitans. 1832 BOWDITCH. Translation of the second volume of the Mecanique Celeste, pages 450 to 455. 1837 BESSEL. Astronomische Nachrichten, Vol. XIV. 1842 BESSEL. Astronomische Nachrichten, Vol. XIX. 1844 BIOT. Traite Elementaire a" Astronomic Physique, Vol. III. 1859 SCHUBERT. Petersburg Memoires, seventh series, Vol. I. 1 86 1 CLARKE. Memoirs of the Royal Astronomical Society, Vol. XXIX. 1866 HERSCHEL. Familiar Lectures on Scientific Subjects. CHAPTER I. . NEWTON. 1. NEARLY two centuries have passed away since the pub- lication of the greatest work known in the history of science. Newton's Philosophic^ Naturalis Principia Mathematica appeared in 1687. The volume is in quarto ; it contains a title-leaf, a dedication to the Royal Society on another leaf, a preface on two pages, some Latin verses by Halley on two pages, then the text consisting apparently of 510 pages, followed by errata on one leaf. I say the text consists apparently of 510 pages ; there are, however, no pages numbered from 384 to 399 inclusive : the third Book begins on page 401, and so perhaps some of this was struck off before the second Book was finished, and a gap was left in the number of pages which proved too large. The second edition of the Principia appeared in 1713, edited by Cotes; the third in 1726, edited by Pemberton. Newton was born in 1642, and died in 1727. 2. Newton's researches on Attractions form Sections xil. and xiii. of the first Book of the Principia. Section xil. con- tains Propositions 70... 84; it relates to the attraction of sphe- rical bodies. Section xiii. contains Propositions 85... 93; it relates to the attraction of bodies which are not spherical. These Sections remain unchanged in the other two editions of the Principia. 3. In his Proposition 70, Newton shews that a particle will be in equilibrium if placed at any point of the hollow part of an indefinitely thin spherical shell, which attracts according to A T. M. A. 1 2 NEWTON. the law of the inverse square of the distance. Newton's de- monstration is remarkable for its simplicity. Let any indefinitely small double cone be described with the position of the attracted particle as vertex ; the areas of the indefinitely small surfaces which the cone intercepts on the shell are ultimately as the squares of the distances of the elements from the vertex : thus the elements exert equal attractions in opposite directions. There- fore the entire shell exerts no action in any direction. We assume here and in the other propositions that the attract- ing body is homogeneous unless the contrary is stated. 4. In his Proposition 71, Newton shews that an indefinitely thin spherical shell attracts an external particle towards the centre of the shell, with a force which varies inversely as the square of the distance of the particle from the centre of the shell. Newton's demonstration is geometrical ; it can, however, be easily translated into an analytical form. Let a be the radius of the shell, c the distance of the particle from the centre of the shell, ds an element of the length of the circle which by revolution round the straight line joining the particle to the centre generates the surface of the shell, r the distance of this element from the particle, y its distance from the axis of revolution. Then the element of surface generated by the revolution of ds is %7ryds ; and the attraction of this element along the axis is - ./ cos 6 ; where k is the thickness of the shell, p is the density, and is the angle between the direction of r and the axis. Let p denote the perpendicular from the centre of the shell on the direction of r. We have p = c sin 6, r* 2rc cos 6 + c 2 = a 2 ; dr re sin 6 hence -^ = clO r c cos 6 ' rpv 27rkpyds n 'Zirkpu cos 9 ardO Thus ^ cos 6 ^2 ^ r r r c cos 6 Zirlcpyadp Zjrkpapdp ~ cr(r-ccos0) ~ cVK-jp 1 ) ' NEWTON. 3 Hence the resultant attraction of the shell will be found by integrating this expression between appropriate limits. If we take and a as the limits of p, we obtain the attraction of either of the two parts into which the shell is divided by the curve of contact of straight lines drawn from the particle to touch the shell; hence these two parts exert equal attractions, and the attraction of the whole shell is pdp which varies inversely as c 2 . The value of the definite integral is a ; and thus the attrac- tion of the whole shell is -- ^ . c We see from this investigation that if any right cone be taken having its vertex at the position of the particle, and its axis coin- cident with the straight line drawn from the particle to the centre of the shell, we can determine the attraction which is exerted by the portion of the shell cut off by the cone : we have only to give an appropriate value to the upper limit of p in the integration. We may observe too that if any indefinitely small cone be taken having its vertex at the position of the particle, the two distinct portions of the shell which it intercepts exert equal attractions. We may observe that Proposition 71 has been very well treated by Professor Thomson: see Cambridge and Dublin Ma" thematical Journal, Vol. in. page 146. 5. Propositions 72... 76 extend the conclusions obtained re- specting indefinitely thin spherical shells to spheres. It appears that Newton arrived at his theorems respecting the attraction of spheres in 1685. See the M&anique Celeste, Vol. v., page 87 ; Rigaud's Historical Essay on the first publication of the Principia, page 27 of the Appendix. 6. Newton's Propositions 77 and 78 relate to the case in which the law of attraction is that of the direct distance. Between Propositions 78 and 79 a Lemma occurs. 12 4 NEWTON. Let x and y be the co-ordinates of a point on a circle ; r the distance of the point from any fixed origin. We have therefore r dr = xdx + ydy = (x - c) dx + ydy Let c be the distance of the centre of the circle from the origin, the centre being on the axis of x. Then (x c) dx -\-ydy = ; therefore rdr = cdx. This result constitutes the Lemma ; it is of course demonstrated geometrically by Newton. Throughout this Chapter we shall translate Newton's geometrical processes into modern mathematical language. 7. In his Proposition 79, Newton finds the attraction of a zone of an indefinitely thin spherical shell on a particle at the centre of the shell. Take the axis of the zone for that of x, and a line at right angles to this through the centre of the shell for the axis of y ; let .a be the radius of the sphere. Then Ztradx represents an element of the zone; and the attraction of this element will be Jcf. ^Tra.-dx, where k denotes the thickness of the shell, and y is a constant which denotes the attraction of a unit of matter, condensed at a point, on a particle at the distance a. Hence the attraction of the zone = hf. 2?r \xdx, the integral being taken between proper limits. If the zone be the segment cut off by the plane x = x^ , we have to integrate between the limits x^ and a. Thus we obtain kfir (a 2 a^ 2 ), that is Jcfiry*, where y l is the radius of the base of the segment. 8. Newton's Proposition 80 investigates the attraction of a sphere on an external particle, the law of attraction being ex- pressed by any function of the distance. Divide the sphere into elements by describing spherical sur- faces indefinitely close to each other from the external particle as centre. Let r be the radius of one of the surfaces of one of the segments of shells thus obtained, and y the radius of the base of the segment ; let <f> (r) denote the law of attraction. Then by Art. 7 NEWTON. we have Trdrfy (r)y* for the attraction of the segment. Let c be the distance of the external particle from the centre of the sphere; then by Art. 6 we have dr = - - : thus the attraction becomes j r ' CTT - <f> (r) y*dx. Hence the resultant attraction of the sphere is r CTT ^ y*dx, where a is the radius of the sphere. J c-a r 9. Newton's Proposition 81 amounts to a transformation of the integral obtained in Art. 8. We have ?/ 2 = a 2 (c x)*, and also y* = r 2 x* ; therefore r 2 = a 2 - c 2 + 2cx. c 2 a 2 Put ~ = 6, and x b x ; thus zc r 2 = 2c (x - b) = 2cx, y* = - 2bc + 2cx -x* = 2cx - (x Hence the resultant attraction <f> (V) dx, r the limits of x' being c a - b and c + a b. As soon as <f> (r) is known we can substitute for r in terms of x, and effect the integration. Newton gives three examples : (1) $(r)=J, (2) (r)-g, (3) $(r) = ^,, where /A in each case is a constant. 10. Newton's Proposition 82 shews that the calculation of the attraction of a sphere on an internal particle may be made to depend on the calculation of the attraction on an external particle. We have found in Art. 8 for the attraction of an element of the sphere irdr <f> (r) y 2 , where r is the distance of the particle from every point of the element. In the same manner Trdr </> (r') ?/ 2 will express the attraction of the corresponding element on another particle which is at the distance r from every point of the element. The two particles and the centre of the sphere HIV 6 NEWTON. of course on the same straight line. Suppose the second particle within the sphere; let c be the distance of the first particle from the centre of the sphere, c' that of the' second, a the radius of the sphere. Let c and c be taken so that cc = cC. In the diagram let *SP = c, SI=c', EF=r, EI=r. As cc = ft* the triangles PSE&ud ESI are similar; thus we have r' = c' r a ' In finding the attraction on the internal particle, we may if we please suppose the matter to be removed which forms a sphere having its centre at the internal particle and radius equal to a c: thus the limits of integration become r = a c' and r =a + c. Suppose </> (?) = ; the attraction on the internal particle ?' i *>*/ the limits being a c and a+ c. Now put - for r ; thus we get TTLL I ,-) ' <lr. and the limits of r are , a and -. + a, that \c J ] r c c is, c, a and c -f a. Hem-e tin- attraction on the internal particle at the distance c I'mm the centre is equal to the product of ( , ] into the attraction V ft / \C / on the external particle at the distance r from the centre. NEWTON. n-1 n ,'\ 2 /V\ A //\ * This is the result which Newton intended to give. He says that the attraction on the particle at / is to the attraction on the particle at P, in ratione composita ex subduplicata ratione distantiarura a centro IS et PS, et subduplicata ratione virium centripetarum, in locis illis P et 7, ad centrum tendentium. It seems to me that instead of P et / we ought to read Jet P. 11. Newton's Propositions 83 and 84 shew briefly that there would be no difficulty in calculating the attraction of a homo- geneous segment of a sphere on a particle situated on the axis of the segment, 12. Newton's Propositions 85, 86, and 87 involve simple general statements, which need not be repeated here. Propositions 88 and 89 shew that if the law of attraction is that of the direct distance, the resultant attraction exerted by a body or a system of bodies is the same as if the body or system were collected at its centre of gravity. 13. Proposition 90 finds the attraction of a circular lamina on a particle which is situated on the straight line drawn through the centre of the lamina at right angles to its plane. Then Propo- sition 91 shews how from this we can deduce the attraction of a solid of revolution on a particle situated at any point of the axis. Newton makes this depend on the problem of finding the area of a certain curve ; that is, in modern language, he leaves only a single integration to be effected. He takes the case of a right cylinder for an example ; and he also states the result for the case of an ellipsoid of revolution, which he calls a spheroid. He shews by a special investigation that a shell bounded by two concentric similar and similarly situated ellipsoidal surfaces of revolution exerts no attraction on a particle placed at any point within the hollow part ; the demonstration is very striking and well known : see Statics, Chapter xni. Of course this result includes Newton's Proposition 70 as a particular case ; but the demonstrations differ and should be carefully compared. 8 NEWTON. Hence follows the important result that along the same radius vector from the centre the attraction of an ellipsoid of revolution on an internal particle varies as the distance from the centre. Newton contented himself with considering ellipsoids of revo- lution; but the processes and results of Proposition 91, as we now know, may be easily extended to ellipsoids which are not solids of revolution. 14. Proposition 92 shews how we may find experimentally the law of attraction of given matter. Form the given matter into such a shape that the resultant attraction can be obtained when the law of attraction is assumed; for example, the shape of a sphere. Then ascertain by experiment what the resultant attrac- tion really is at various distances ; and thus we shall be guided in assuming a law of attraction and verifying the assumption. 15. Proposition 93 treats of the attraction of an infinite plane lamina, deducing it from Proposition 90. A scholium to this Proposition gives some interesting remarks relating to the motion of a particle acted on by a force the direction of which is always parallel to a fixed straight line. 16. Newton's Propositions on Attractions are illustrated by a good commentary in the edition of the Principia which is known as the Jesuits' edition. They had been previously discussed by Maupertuis, as we shall see in another Chapter. Notes by Plana on some of the Propositions will be found in the Memorie della Eeale Accademia. . .di Torino, second series, Vol. XL, 1851. 17. We pass now to the investigations made by Newton with respect to the Figure of the Earth ; they are contained in Propo- sitions XVIII., XIX., and XX. of the third Book of the Principia : these Propositions remain substantially the same in the second and third editions as in the first, but modifications occur arising from additional information as to the facts involved. Before we consider these Propositions we ought to advert to Newton's remarkable conjecture which is contained in Proposition X. Newton here suggests that the mean density of the Earth may NEWTON. 9 be five or six times that of water :...verisimile est quod copia materiae totius in Teml quasi quintuple vel sextuplo major sit quam si tota ex aqua constaret. We may now consider it certain that the mean density is between five and six times that of water. Laplace draws attention to Newton's remarkable conjecture in the Connaissance des Terns for 1823, page 328. It will be convenient to give the enunciations of Newton's Propositions XVIII., XIX. and XX. XVIII. Axes Planetarum quse ad eosdem axes normaliter du- cuntur minores esse. XIX. Invenire proportionem axis Planetse ad diametros eidem perpendiculares. XX. Invenire et inter se comparare pondera corporum in Teme hujus regionibus diversis. 18. Proposition XVIII. contains a general statement that the planets are not accurately spherical. In the first edition Cassini and Flamsteed are quoted as authorities for this statement with respect to Jupiter ; in the second edition instead of these names we are referred to astronomers in general. 19. Proposition XIX. undertakes to determine the ratio of the axes of a planet. This important process consists of various steps. In the first edition Newton begins by saying briefly he finds from calculation that the centrifugal force at the equator is to the force of attraction there as 1 to 290. In the second edition the details of the calculation are supplied, and the ratio obtained is that of 1 to 289 : this ratio is that which is now usually given in our elementary books, and it will be convenient to adopt it as we proceed with an account of Newton's investi- gation. Suppose two slender canals of homogeneous fluid, one along the polar radius of the earth, and the other along an equatorial radius. The resultant attraction on the equatorial canal must be greater than that on the polar canal in the ratio of 289 to 288 in order that there may be relative equilibrium. For in proceeding along any given radius inside the earth the attraction varies as 10 NEWTON. the distance, and the centrifugal force varies as the distance ; hence the ratio of the latter to the former is constant along the equatorial radius ; so that the effect of the centrifugal force may be considered equivalent to removing - of the force of attraction. 20. Newton's next step is to compare the attraction of an oblate ellipsoid of revolution on a particle at its pole with the attraction of the same body on a particle at its equator, the ellip- ticity being supposed very small. He states his results without giving his process at full. It will be remembered that he had found an expression for the attraction of an ellipsoid of revolution at any point of its axis : see Art. 13. I. Suppose an oblate ellipsoid of revolution formed from an ellipse, such that the major semi-axis CA is to the minor semi-axis CQ as 101 is to 100. The reader can easily draw the diagram for himself. Newton says that the attraction at Q would be to the attraction of a sphere having C for centre and CQ for radius, as 126 is to 125. If denote the ellipticity we know from our 4e modern works that this ratio is that of 1 + to 1 ; see Stohca, o Chapter xili. ; this agrees closely with Newton's numerical ex- ample. II. Suppose a prolate ellipsoid of revolution formed from the same ellipse. Newton says that the attraction at A would be to the attraction of a sphere having G for centre and CA for radius, as 125 is to 126. If e denote the ellipticity we know from our 4e modern works that this ratio is that of 1 =- to 1 ; see Statics, o Chapter XIII. : this agrees with Newton's numerical example. 2 In the first edition Newton put the fraction after 126 and 15 125 in I. and II. The fraction was removed by Cotes : see the Correspondence of Newton and Cotes, page 69. III. Now return to the oblate ellipsoid of revolution. Suppose a particle at A: Newton says that the attraction on it will be a mean proportional between the attractions of the sphere and of the NEWTON. 11 prolate ellipsoid of revolution in II. We will develop his argu- ment. Begin with the sphere having CA for radius ; if we change the radius which lies along CQ into CQ we deduce the oblate ellipsoid of revolution ; if in this we change the radius which is at right angles to CA and CQ into a radius equal to CQ, we deduce the prolate ellipsoid of revolution. Now each of these changes may be assumed to have affected the attraction to the same amount ; and so the attraction of the oblate ellipsoid of revolution is approximately an arithmetical mean between the attractions of the sphere and of the prolate ellipsoid of revolution. Moreover the arithmetical mean between two nearly equal quantities is practically equivalent to the geometrical mean. Hence, finally, the attraction of the sphere with centre C and radius CA is to the attraction of the oblate ellipsoid of revolution on the particle at A as 126 is to 125J. IV. Thus we have Attraction of oblate ellipsoid of revolution at the pole 126 = .j-^ x attraction of sphere of radius 100 at its surface ; attraction of sphere of radius 101 at its surface 126 = -ia-i x attraction of oblate ellipsoid of revolution at its equator ; attraction of sphere of radius 100 at its surface 100 x attraction of sphere of radius 101 at its surface. Hence we find by multiplication that the ratio of the attraction of the oblate spheroid of revolution at the pole to the attraction at the equator is expressed by x x , that is, by nearly. 21. In future I shall use the single word oblatum instead of oblate ellipsoid of revolution, and the single word oblongum instead of prolate ellipsoid of revolution. 22. I shall now make some remarks on the statement by Newton which forms the paragraph III. of Art, 20. 12 NEWTON. If we cut the three solids by two adjacent planes at right angles to A G we obtain slices, two of which are circular and the other elliptical. It is not difficult to believe that in passing from the larger circular slice to the elliptical slice we diminish the attraction by the same amount as we do in passing from the elliptical slice to the smaller circular slice. In fact, the decrement of mass is about the same in the two cases, and the mass lost is at about the same situation with respect to the attracted particle in the two cases. It is easy to test the statement by the aid of the modern formula ; see Statics, Chapter xm. First take an oblatum of density unity ; let r be its greatest radius and r\/(l e 2 ) its least radius. The attraction at the equator ir = 27rr (1 - e 2 ) * sin 3 Q (1 - e 2 sin 2 0)~ l dd Next take an oblongum ; let r be its greatest radius and r V(l 2 ) its least radius. The attraction at the pole 47rr (1 - e 2 ) P sin cos 2 d (1 - e 2 cos 2 6}~ l dO The attraction of a sphere of radius r at its surface = Suppose e so small that we may reject e* and higher powers A-frf / f?\ of e ; then the first of these attractions reduces to f 1 ^ J , 4-Trr / 2e 2 \ and the second to ^- (1 -- ^ J , so that the first is an arithmetical mean between the second and the third. But this statement does not hold if we carry our approximations as far as e 4 inclusive ; it will be found then that the first of the attractions is rather better represented by the mean proportional between the second and the third than by their arithmetical mean. NEWTON. 13 It will be convenient to quote Newton's own words, premising that in his diagram PCQ is the polar diameter and ACB an equatorial diameter. Est autem gravitas in loco A in Terram, media proportionalis inter gravitates in dictam Sphseroidem et Sphseram, propterea quod Sphaera, diminuendo diametrum PQ in ratione 101 ad 100, vertitur in fignram Terra? ; et hsec figura diminuendo in eadern ratione diametrum tertiam, quse diametris duabus AJ3, PQ perpendicularis est, vertitur in dictam Spha3roidem, et gravitas in A, in casu utroque, diminuitur ia eadem ratione quam proxime. The words in eadem ratione, which occur at the end of this extract, seem to have been formerly misunderstood ; it was sup- posed Newton intended to affirm that the attractions of the three bodies were in the same ratio as their volumes. But this is not the case. The volume of the oblatum is *J(I e*) ; the volume of the oblongum is 77 (1 -? e 2 ) ; and the volume of the o sphere is ^- : these volumes are not in the ratio of the attrac- ts tions exactly nor approximately to the order of e 2 . Hence the following passage, which occurs in a note in the Jesuits' edition of the Principia, is erroneous: " ...... attractiones sphserse, sphasroidis compressae, et sphaeroidis oblongatae, sunt respective ut quantitates materise in illis corporibus contentaa quam proxime." The words in eadem ratione, which occur at the end of the extract from Newton, must be understood to mean only to the same amount ; and must not be taken in exactly the same sense as in the middle of. the passage. It is obvious, however, that it would have been simpler and more natural to say, that the attraction of the oblatum is an arithmetical mean between those of the sphere and the oblongum, than to say that it is a mean proportional between them, to the order of accuracy which Newton adopts. 23. Newton now compares the resultant attraction on the fluid in a slender canal having AC for its axis, with that on the fluid in a slender canal having QC for its axis. He finds that 14 NEWTON. these resultants are in a ratio compounded of the ratios of the lengths and of the ratios of the attractions at the extremities A and Q : see Art. 33. Thus the resultant attraction on the fluid in the canal AC is to that on the fluid in the canal QC, as 101 x 500 is to 100 x 501, that is, as 505 is to 501. Hence Newton infers, that if the centrifugal force at any point of A C is to the attraction at that point as 4 is to 505, the weights of the fluids in the two canals will be equal, and the canals in relative equilibrium. 24. The last step in the preceding Article is more obvious to us, who have the modern theory of the equilibrium of fluids, than it would have been before that theory was constructed. We see that in the state of relative equilibrium the pressure at C must be the same in every direction round C: the pressure on a given area at right angles to A G will be measured by the weight of a column of fluid having that area for its section ; and similarly for the pressure on an area at right angles to QC. The canals of fluid which Newton considered were rectilinear, meeting at the centre. Other writers, especially Clairaut, con- sidered canals of various forms, curvilinear as well as rectilinear, meeting at any point of the body. The more simple case to which Newton restricted himself may be conveniently described as that of central columns; so that the word canal may be in future used in Clairaut's more general sense. 25. It is now necessary to explain carefully the sense in which the words attraction, gravity and weight will be used in this history. By the attraction of the Earth at any point, I understand that force which the Earth would exert, supposing it did not rotate on its axis. By gravity I denote the force which arises from the com- bination of the attraction and the so-called centrifugal force ; and weight may be considered as an effect produced by gravity as the cause. As we may measure the cause by the effect, it will be found that it is often indifferent whether we use the word gravity or the word weight : but it is convenient to have both words at our service. The word weight is thus left in its ordinary sense NEWTON. 15 as denoting an effect which is actually produced in the existing constitution of things, and actually observed. It would be convenient if we had a word for the effect which corresponds to attraction as a cause ; but such a word is not very often required, because practically we are not concerned with an Earth at rest, but with one which rotates. For want of such a word I employ the phrase resultant attraction in Arts. 19 and 23. The distinctions which we have here drawn actually exist, and various modes have been adopted for preventing confusion. Some writers, indeed, leave us to gather from the context the sense in which they use their terms. This is the case with Newton himself. Thus, for instance, in the passage quoted in Art. 22, he uses gravitas for what I call attraction. In his Propo- sition XX. he uses gravitas for what I call gravity. In his Propo- sition XIX. he uses pondus sometimes to express the effect pro- duced by what I call attraction, and sometimes in the sense I give to weight. To secure accuracy I have used not his words but my own. Maupertuis used gruvitd for my attraction, and pesanteur for my gravity; and Clairaut followed Maupertuis. See Mauper- tuis's Figure de la Terre, page 158, and Clairaut's Figure de la Terre, page xiii. In the Mecanique Ce'leste, Vol. v. page 2, we have the same use of gravite. Maupertuis had previously used pesanteur re'duite for my gravity: see the Paris Me'moires for 1734, page 97. Bouguer used pesanteur primitive for my attraction, and pesan- teur actuelle for my gravity: see the Paris Mtmoires for 1734, page 22, and Bouguer's Figure de la Terre, page 169. Maclaurin used gravitation as Maupertuis used pesanteur, and gravity as Maupertuis used gravite: so also did Thomas Simpson. See Maclaurin's Fluxions, page 551, and Simpson's Mathematical Dissertations, page 22. The word gravitation has been employed by some eminent modern writers in about the same sense as my attraction; as, for example, in Airy's article on Gravitation in the Penny Cyclopaedia, and by Thomson and Tait in their Natural Philosophy, Vol. I. page 167. 16 NEWTON. Boscovich used gravitas primitiva for my attraction, and gravitas residua for my gravity : see his De Litteraria Expedi- tions, page 403. But in the Bologna Commentarii, Vol. IV. page 382, he uses tota gravitas for the gravitas residua of his book. In the French translation of Boscovich's book, we have gravite primitive for his gravitas primitiva, and gravite alsolue for his gravitas residua ; see the pages 8 and 384 of the translation. The word pesanteur is used by Bailly in his Histoire de VAs- tronomie Moderne, Vol. ill. page 4, as equivalent to my attraction ; but in general the sense assigned by Maupertuis and Clairaut to pesanteur has been adopted by their successors. In French the word poids is almost equivalent to my weight; see Maupertuis's Figure de la Terre, page 155. 26. We now return to Art. 23. The result there obtained is, that if the ellipticity be r- - , then for relative equilibrium the centrifugal force at the equator must be - - of the attraction there. Newton now forms a proportion. He says : Verum vis centrifuga partis cuj usque est ad pondus ejusdem lit 1 et 289, hoc est, vis centrifuga, quse deberet esse ponderis pars 4 1 ? est tantum pars ^ QQ . Et propterea dico, secundum Regulam OUO Zoy auream, quod si vis centrifuga ^7^ faciat ut altitude aquae in crure ACca OUD superet altitudinem aquse in crure QCcq parte centesima totius altitudinis : vis centrifnga - - faciet ut excessus altitudinis in crnre ACca sit alti- 2O9 tudinis in crure altero QCcq pars tantum . mA9 These are the numbers of the second and third editions ; in 1 13 the first edition Newton has instead of , and -- in- stead of 5SQ. In his diagram ca is parallel and adjacent to 2Z\) CA, and cq is parallel and adjacent to CQ. NEWTON. 17 27. If we put Newton's investigation into a modern form it will stand thus. Let e be the ellipticity, supposed very small ; then the attraction of an oblatum at its pole is to the attraction of the oblatum at its equator as 1 + ^ is to 1 ; see Statics, Chapter xin. Hence, as in Art. 23, the ratio of the resultant attraction on the fluid in A C to the resultant attraction on the fluid in Q C 1 -f e 4e is expressed by , that is, by 1 + approximately. There- 1+ l fore for relative equilibrium we must have the centrifugal force 4e at any point in AC equal to -=- of the attraction at that point. Newton, in fact, sees that the fraction which we have found fc 5 4e to be must be proportional to e; hence it may be denoted by ke, where k is some constant. Then, when e = ^- , he finds 4 404 4 that ke ^^ , so that k = -^-= = -= . 50o 505 5 28. The result obtained by Newton then is that if the Earth is homogeneous and its shape the same as if it were entirely fluid, the ellipticity must be ~^r , supposing it to be very small ; that is, the ellipticity must be 7 of the ratio of the centrifugal force to the 4) attraction at the equator. The result is very important in the theory of the subject ; but we know now that the ellipticity is about JT- , and we are confident that the Earth is not homogeneous. 29. Newton proceeds to some remarks on the oblateness of Jupiter. Let j denote the ratio of the centrifugal force to the attraction at the equator, and e the ellipticity ; then we have shewn that . 4e ?-t' T. M. A. 2 18 NEWTON. In his first edition Newton erroneously asserts thatj* is inde- pendent of the density. He says : Si Planeta vel major sit vel densior, minorve aut rarior quam Terra, manente tempore periodico revolutionis diurnse, manebit proportio vis centrifugse ad gravitatem, et propterea manebit etiam proportio dia- metri inter polos ad diametrum secundum sequatorem. Accordingly he considers j to vary inversely as the square of the time of rotation ; so that the value of j for Jupiter becomes 29 - times its value for the Earth : and hence Jupiter's ellipticity 5 29 is taken to be times that of the Earth, so that the ratio of the o difference of the axes to the minor axis is about - . In the second edition Newton corrects his error. He says : Si Planeta major sit vel minor quam Terra manente ejus densitate ac tempore periodico revolutionis diurnse, manebit proportio vis centri- fugae ad gravitatem . . . Accordingly he now rightly considers j to vary inversely as the density as well as inversely as the square of the time of rotation ; so that, taking the density of Jupiter to be - of the o density of the Earth, the ratio of the difference of the axes to the 29 5 1 1 minor axis becomes - x - x , that is, about . O JL 944 In the third edition, the density of Jupiter is taken as ' * of 400 the density of the Earth ; and the ratio of the difference of the axes to the minor axis becomes about . y i 30. In the first and second editions these words occur at the end of Proposition XIX. : Haec ita se habent ex Hypothesi quod uniformis sit Planetarum materia. Nam si materia densior sit ad centrum quam ad circumfereii- tiam, diameter, quse ab oriente in occidentem ducitur, erit adhuc major. NEWTON. 19* Thus Newton considered that if the Earth, instead of being of uniform density, were denser towards the centre than towards the surface, the ellipticity would be increased ; see also Art. 37. But Newton was wrong. Assuming the original fluidity of the Earth, the ellipticity is diminished by increasing the density of the central part, supposed spherical, and making it solid. This was shewn by Clairaut, who pointed out Newton's error: see Clairaut's Figure de la Terre, pages 157, 223, 224, 253... 256. Clairaut, however, ought to have remarked that Newton omitted the passage in his third edition. In his third edition, as I have just said, Newton omitted the above passage. He says instead : Hoc ita se babet ex hypothesi quod corpus Jovis sit uniformiter densum. At si corpus ejus sit densius versus planum sequatoris quam versus polos, diametri ejus possunt esse ad invicem ut 12 ad 11, vel 13 ad 12, vel forte 14 ad 13. Then, after stating some observations as to the ratio of the axes of Jupiter, Newton says : Congruit igitur tlieoria cum phsenomenis. Nam planetae magis in- calescunt ad lucem Solis versus sequatores suos, et propterea paulo magis ibi decoquuntur quam versus polos. It might then appear that in his third edition Newton had recognised his error ; but we shall find, when we discuss Propo- sition XX., that a distinct trace of the error still remains : see Arts. 38 and 41. 31. I do not feel certain as to the meaning of the sentence "congruit... polos," which I have quoted in the preceding Article. Since heat expands bodies it would appear that the equatorial parts ought by the Sun's action to be rendered less dense than the polar parts. The same difficulty presented itself to Bosco- vich: see page 475 of his De Litteraria Expeditione, and pages 89 and 380 of Stay's Philosophies Recentioris, Vol. II. Clairaut says in his Figure de la Terre, pages 223, 224 : De la meme mauiere, on voit combien il etait inutile a M. Newton, lorsque sa theoric lui donuait pour Jupiter, une ellipticite moindre que 22 20 NEWTON. celle qui resulte des ohservations, d'aller imagincr que Toquatonr de eette planete otant continuellomont expose aux ardours du soleil, etait plus dense quc le roste de la planoto. II n'avait qu'a supposer siniple- nieut quo le noyau etait plus dense que le roste de la planetc... Tlie word inoindre in tliis passage is wrong; for Newton's theoretical value of Jupiter's ellipticity in the third edition is greater than the value in the observations he quotes. 3'2. I will now briefly indicate the changes which Newton's Proposition XIX. underwent in the later editions with respect to the facts which it involves. In the first edition, Newton takes the mean semi-diameter of the Earth to be UMJloSOO Paris feet, jtucta nupcrain Gallurum men- sura in : this alludes to Pi card's measurement of the length of an arc of the meridian in France. In the second edition, Newton refers to the measurements made by Picanl, by Norwood, and by Cassini ; according to Cassini's measurement, the semi-diameter is l%'9553i) Paris feet, supposing the Karth spherical. This Cassini is the first of the dis- tinguished family ; his name was Jean Dominique Cassini, but he is often called simply Dominique Cassini. In the third edition, Newton refers also to the measurement made by the son of Dominique Cassini, who is known as Jacques Cassini. Picanl had obtained 57060 toises for the length of a degree ; the arc measured by D. and J. Cassini gave 570G1 toises for the mean length of a degree. We shall see as we pro- ceed with the history that these measurements were subsequently re-examined and corrected. !}*}. \Ve now come to Newton's Proposition XX., the object of whirli is to compare the weights of a given body at different places on the Earth's surface. Newton begins with an important result, which is deduced from his principle of balancing columns: see Art. 23. At any point of the Karth's surface, let . f denote the force of gravity, refill'/'!/ ultHit/ t]t( } r<n/iuf<; let r be the distance of this point from NEWTON. 21 the centre ; let x be the distance from the centre of any point on the same radius : then the force of gravity at this point resolved along the radius will be , because along the same radius within the Earth both the resolved attraction and the resolved centri- fugal force vary as the distance. Hence the resolved weight of a column of fluid extending from the surface to the centre, is rx 1 -dx, that is, by ^fr. And as the resolved r 2 weight of every column must be the same, for relative equilibrium, we must have^r constant, and so /must vary inversely as r. This is equivalent to an expansion of Newton's biief outline : it becomes more obvious to modern readers by the aid of the theory of the equilibrium of fluids. It is clear that the final result would be true if the resolved force within the Earth varied as any direct power of the distance instead of as the first power. Thus we may say that the weight of a given body at any point of the Earth's surface when resolved along the radius varies inversely as the radius. Newton, however, omits the words which I have printed in Italics. Since the Earth is very nearly a sphere, the omission will be of no consequence practically, but theoretically it is important to be accurate. 34. Newton proceeds thus : Unde tale confit Theorema, quod incrementum ponderis, per- gendo ab ^Equatore ad Polos, sit quain proxime ut Sinus versus latitudinis duplicatae, vel quod perinde est ut quadratum Sinus recti Latitudinis. The result here stated may be thus investigated. Let g denote the weight of a given body at any point of the Earth's surface ; let r denote the radius at that point, and <f> the angle which the normal at that point makes with the radius. Then, assuming that the direction of gravity coincides with the normal, the resolved part of the weight along the radius will be gcos(f>. Therefore, by Art. 33, we have g cos < = - > where \ is some constant, so that 22 NEWTON. q = 7-. Let G denote the weight of the given body at the rcos<f> equator, a the radius of the equator ; then r cos 1 -1). os cz> / By neglecting powers of the ellipticity beyond the first, it is found that varies as the square of the sine of the rcoscj) a latitude. The words printed in Italics in the above investigation involve a principle which is familiar to us in the modern theory of fluids ; as we shall see hereafter, this principle was used by Huygens. Newton, however, tacitly assumes that the direction of gravity coincides with the radius. It is true, that to the order of approxi- mation which we adopt cos <f> may be put equal to unity, and so there is no practical error involved in Newton's assumption : but theoretically his investigation of the very important proposition now before us is thus rendered obscure and imperfect. 35. In Newton's second and third editions, we have after the passage last quoted these words : " Et in eadem circiter ratione augentur arcus graduum Latitudinis in Meridiano." This is a fact in the theory of the Conic Sections with which we are now familiar. Let p denote the perpendicular from the centre of an ellipse on the tangent at any point ; then the radius of cur- vature varies as -= . Thus the increase of the radius of curvature P in proceeding from the equator towards the pole varies as 3 3 ; and by neglecting powers of the ellipticity beyond the first it is found that this varies as the square of the sine of the latitude. 36. In the first edition Newton calculated the relative weights of a given body at Paris, Goree, Cayenne, and the Equator. As observations of the length of a seconds pendulum had been made at Paris, Goree, and Cayenne, the relative weights of a body at those places were known, and thus a test of the accuracy of the theory was furnished. NEWTON. 23 37. The following sentences occur in the first edition ; they are repeated substantially in the second edition, but are omitted in the third edition. Haec omnia ita se habebuut, ex Hypothesi quod Terra ex uniformi materia constat. Nam si materia ad centrum paulo densior sit quam ad superficiem, excessus illi erunt paulo majores ; propterea quod, si materia ad centrum redundans, qua densitas ibi major redditur, sub- ducatur et seorsim spectetur, gravitas in Terram reliquam uniformiter densani erit reciproce ut distaritia ponderis a centro ; in materiam verb reduiidantem reciproce ut quadratum distantiae a materia ilia quam proxime. Gravitas igitur sub sequatore minor erit in materiam illam redundantem quam pro computo superiore, et propterea Terra ibi propter defectum gravitatis paulo altius ascendet quam in precedentibus definitum est. The preceding sentences contain a portion of truth. Suppose that a mass of rotatory homogeneous fluid has taken the form which Newton assigns for relative equilibrium. Then gravity at the pole is to gravity at the equator, inversely as the corre- sponding distances from the centre ; and if we suppose the ob- latum to become solid, this ratio is not changed. Next suppose that the central part is made denser than the rest, this central part being spherical in shape. Thus the gravity is increased both at the pole and at the equator ; but the additional gravity at the pole is to that at the equator inversely as the squares of the corresponding distances. Therefore the whole gravity at the equator bears to the whole gravity at the pole a less ratio than for the case of the homogeneous body. But now Newton in some way returns, as it were, to the sup- position of fluidity. It is not obvious whether the whole mass is supposed to be fluid, or whether the central spherical part is still left solid. In either case a new investigation would have to be supplied, in order to determine the figure of the fluid part for relative equilibrium; and no use could be made of a result obtained from the balancing at the centre of homogeneous, columns. As we have said in Art. 30, the investigations of Clairaut bring out the ellipticity less than for the homogeneous case, aud not greater as Newton stated. 24 NEWTON. According to Clairaut, Newton's error lay in thinking that gravity at the ends of the columns must be inversely proportional to the lengths of the columns for relative equilibrium, whether the fluid is homogeneous or not. See Clairaut's Figure de la Terre, pages 224 and 256, and Stay's Philosophice JRecentioris, Vol. n. page 370. 38. In the first edition, as we have stated, Newton referred to pendulum observations at only three places, Paris, Goree, and Cayenne. These observations indicated a rather greater varia- tion in the length of the seconds pendulum than the theory suggested. So Newton says: et propterea (si crassis hisce Observationibus satis confidendurn sit) Terra aliquanto altior erit sub sequatore quam pro superiore calculo, et densior ad centrum quam in fodinis prope superficiem. He then points out the advantage which would be derived from a set of experiments for determining the relative weights of a given body at various places on the Earth's surface. 39. In the second edition Newton gave a table of the lengths of a degree of the meridian and of the lengths of the seconds pendulum in different latitudes. This table was computed by the aid of his theory, taking from observation the length of a degree of the meridian in the latitude of Paris, and also the length of the seconds pendulum there. The table is repeated in the third edition; but the lengths of the degrees are not the same as in the second edition. The lengths are expressed in toises ; at the equator, at 45, and at the pole, the lengths are respectively 56.909, 57283, and 57657 in the second edition ; while in the third edition they are 56637, 57010, and 57382: the difference is, of course, owing to the adoption of a fresh result from the measurement in France. After the table in the second edition we have these words : Constat autem per hanc Tabulam, quod graduum insequalitas tarn parva sit, ut in rebus Geographicis figura Teroe pro Sphaerica haberi possit, quodque insequalitas diametrorum Terrse facilius et certius per NEWTON. 25 experimenta penduloram deprehendi possit vel etiara per Eclipses Lunse, quam per arcus Geographice mensuratos in Meridiano. I cannot understand how the ratio of the axes could be found by pendulum experiments or by eclipses better than by measured arcs. In the third edition the words which follow "haberi possit " are omitted, and instead of them we have "praBsertim si Terra paulb densior sit versus planum sequatoris quam versus polos." 40. In the second and third editions Newton referred to many more pendulum observations than in the first edition. We see from pages 69... 89 of the Correspondence of Newton and Cotes, that the arrangement of this part of the work for the second edition was a matter of some trouble. The figures in the final draft of Newton were corrected by Cotes : compare pages 85 and 92 of the Correspondence with the second edition of the Prin- cipia. The facts are stated nearly in the same terms in the second and third editions. 41. The more numerous observations to which Newton could now appeal, concurred with the smaller number before used, in giving a greater variation to the length of the seconds pendulum than theory suggested. Accordingly, the sentence which we quoted in Art. 38, appears in the second and third editions, omitting the words si crassis...sit. Newton adds, however, "nisi forte calores in Zona torrida longitudinem Pendulorum aliquan- tulum auxerint." The supposition that the pendulum observations required a greater ellipticity than ~^. was shewn to be untenable by Clairaut: see his Figure de la Terre, page 252. 42. In the second edition Newton seems to come to the conclusion that we should correct the theory by observation, and thus take 31^ miles as the excess of the equatorial semi-diameter over the polar semi-diameter. In the third edition, however, he seems to consider that we may hold to the value, 17 miles, fur- nished by theory. 43. In the second edition Proposition XX. ends with a para- graph in which Newton adverts to the hypothesis, founded on 26 NEWTON. some measurements by Cassini, that the Earth is an oblongum : Newton deduces results from this hypothesis which are contrary to observations. The paragraph does not appear in the third edition, although the oblong form continued to find advocates for some years after the death of Newton. 44. Newton's investigations in the theories of Attraction and of the Figure of the Earth may justly be considered worthy of his great name. The propositions on Attraction are numerous, exact, and beautiful ; they reveal his ample mathematical power. The treatment of the Figure of the Earth is, however, still more striking ; inasmuch as the successful solution of a difficult problem in natural philosophy is much rarer than profound researches in abstract mathematics. Newton's solution was not perfect ; but it was a bold outline, in the main correct, which succeeding investi- gators have filled up but have not cancelled. Newton did not demonstrate that an oblatum is a possible form of relative equi- librium ; but, assuming it to be such, he calculated the ratio of the axes. This assumption may be called Newton's postulate with respect to the Figure of the Earth : the defect thus existing in his process was supplied about fifty years later by Stirling and Clairaut. The difficulty arose from the imperfect state of the theory of fluid equilibrium, which undoubtedly must have pro- duced many obstacles for the earliest investigators in mixed mathematics. Clairaut subsequently gave methods which are sound and satisfactory to a reader who can translate them into modern language ; but even these may have appeared obscure to Clairaut's contemporaries. Euler, in the Berlin Memoires for 1755, first rendered Hydrostatics easily intelligible by introducing a symbol p to measure the pressure at any point of a fluid. 45. Besides the defect in Newton's theory which we have pointed out, Laplace finds another, saying in the Me'canique Celeste, Vol. V. page 5, " II suppose encore, sans demonstration, que la pesanteur a la surface, augmente de 1'equateur aux pdles, comme le carre' du sinus de la latitude." But Laplace is not right. Newton did not absolutely assume the proposition ; he gave a demonstration though it was imperfect; see Art. 34. NEWTON. 27 Laplace's language is inaccurate moreover ; it is not gravity that increases as the square of the sine of the latitude, but the varia- tion in gravity. Laplace proceeds to observe that Newton re- garded the Earth as homogeneous, while observations prove incontestably that the densities of the strata increase from the surface to the centre. Laplace's language could scarcely be stronger if borings had actually been executed from the surface to the centre, and had thus rendered the strata open to inspec- tion. He means, of course, that by combining observations made at various places on the surface of the Earth with the suggestions of theory, we are led to infer that the density increases from the surface to the centre. See the Mecanique Celeste, Vol. v. page 12. Laplace truly says that, notwithstanding the imperfections, the first step thus made by Newton in the theory must appear immense. 46. A notice of the Principia was given in the Philosophical Transactions, Vol. xvi. 1687, I presume by Halley, who was then Secretary of the Society ; the simple but expressive words we find on page 297 are still as applicable as they were then : and it may be justly said, that so many and so Valuable Philosophical Truths, as are herein discovered and put past dispute, were never yet owing to the Capacity and Industry of any one Man. CHAPTER II. HUYGENS. 47. WE have now to examine an essay by Huygens entitled Discmirs de la Cause de la Pesanteur. A. small quarto volume was published at Ley den, in 1690, entitled Traite* de la Lumiere...Par C.H.D.Z. Avec un Discours de la Cause de la Pesanteur. The letters C.H.D.Z. stand for Christian Huygens de Zulichem. The volume consists of two parts. Pages 1...124 relate to Light ; they are preceded by a Preface, and a Table of Contents, on 6 pages, which belong to this part of the volume. After page 124 is a Title-leaf for the part relating to Weight ; then a preface on pages 125... 128; then a leaf containing a Table of Contents; and then the text on pages 129... 180. 48. We of course pass over the part relating to Light, merely remarking that it is memorable as laying the foundation of the Undulatory Theory. The part relating to Weight is said to appear in a Latin version in the Opera Peliqua of Huygens : hence it is sometimes cited by a Latin title, De causa gravitatis, or De vi gravitatis. My references will all be made to the original edition in French, pub- lished during the author's lifetime. 49. The last paragraph of the Preface gives information as to the date of composition : La plus grande par tie de ce Discours a este ecrite du temps que je demeurois a Paris, et elle est dans les Registres de 1'Academie Royale HUYGENS. 29 des Sciences, jusques a 1'endroit ou il est parle de Talteration des Pen- dnles par le mouvement de la Terre. Le reste a este adjoute plusieurs annees apres : et en suite encore 1' Addition, a 1'occasion qu'on y trou- vera indiquee au commencement. The former part of the Discourse, which we are here told had been written many years since, is of no value. 50. The theory of Huygens to account for Weight is ex- pounded on pages 129... 144 of the work; we may say briefly that this theory is utterly worthless. Huygens assumes the ex- istence of a very rare medium moving about the Earth with great velocity, not always in the same direction. This rare matter is surrounded by other bodies, and so prevented from escaping; and it pushes towards the Earth any bodies which it meets. This vortex has passed away, as well as those similar but more famous delusions with which the name of Des Cartes is connected. 51. Two incidental matters of some interest may be noticed, On his page 138, Huygens says that there is an invisible pon- derable matter present even in the space from which air has been exhausted : so that it would appear he took the partial exhaus- tion produced by an air-pump for complete exhaustion. On his page 141, he starts a difficulty which we now know has been removed by experiments : ...De plus, en portant un corps pesant au fond d'un puits, ou dans quelque carriere ou mine profonde, il y devroit perdre beaucoup de sa pesanteur. Mais on n'a pas trouve', que je scache, par experience qu'il en perde quoy que ce soit. 52. The really valuable part of the Discourse commences on page 145. Huygens says that at Cayenne the seconds pendulum had been found to be shorter than at Paris. As soon as he heard this, he attributed it to the rotation of the Earth. Accordingly he gives a very good explanation ; assuming that there is at the surface of the Earth a force of constant magnitude directed to- wards the centre, and that there is also a centrifugal force. He shews by calculation, that the centrifugal force at the equator 30 HUYGENS. is about r of the central force. He calculates that the seconds pendulum at Cayenne should be ^ of a line shorter than at Paris ; Richer made.it 1J lines shorter by observation. Huygens calculates that the plumb-line at Paris deviates nearly 6 minutes from the position it would take if there were no centrifugal force. 53. On his page 152, Huygens states a principle which has since generally been called by his name ; he says the surface of the sea is such that at every point the direction of the plumb-line is perpendicular to the surface. The principle may be stated more generally thus : the direction of the resultant force at any point of the free surface of a fluid in equilibrium must be normal to the surface at that point. 54. On page 152, we arrive at the Addition to which Huygens referred in his preface : see Art. 49. His attention was turned to the subject again by examining an account of some more pendulum experiments, and by reading Newton's Principia. Huygens first calculates the ratio of the axes of the Earth. He adopts Newton's principle of the balancing of the polar and equa- torial columns ; but retains his own hypothesis, that the attractive force is central and constant at all distances. Thus he makes the ratio of the axes to be that of 577 to 578. 55. Huygens next finds the equation to the generating curve of the Earth's surface. He considers it difficult to use his own principle of the plumb-line, stated in Art. 53 ; and so he uses the principle of balancing columns. He extends this principle beyond the application which Newton made of it : see Art. 24. Huygens contemplates canals of various forms, not necessarily passing through the centre. He says on his page 156: "et mesme, cela doit arriver de quelque maniere qu'on congoive que le canal soit fait, pourvu qu'il aboutisse de part et d'autre a la surface." Let the constant force be denoted by X, the angular velocity by &>, and the equatorial radius by a ; take the axis of x to coin- HUYGENS. 31 cide with the polar diameter, and the axis of y with an equatorial diameter. Then, by modern methods, we find for the equation to the curve which by revolution generates the surface of the Earth -A' ..................... CD- This coincides with Huygens's result. We may deduce the ratio of the axes from (1) ; we shall thus get the same value as Huygens obtained before he investigated the equation to the curve. Put y in (1), thus : x a(\ r- ) ; therefore the ratio of the axes is that of 1 ^ ^r to 1, that is, the ratio of 577 to 578. If 6 and j have the same meaning as in Art. 29, we see that Huygens's result may be expressed thus: e = ^. 56. Huygens says on his page 159, that even if we do not suppose the central force to be constant, his result remains almost unchanged. It is important to demonstrate this : and we shall accordingly shew that the result is approximately true, whatever may be the law of the force, which is assumed to be central. Let </> (r) denote the force at the distance r from the centre. Then, by modern methods, we find for the equation to the gene- rating curve * 22 (j> (r) dr ~ = constant. Let a denote the equatorial radius, and b the polar. By putting y = 0, we determine the value of the constant, and the equation becomes Now put y = a ; thus 32 HUYGENS. this is an analytical expression of Newton's principle of the balancing of central columns. We may put the expression in the form If a b is very small this gives approximately o) 2 a 2 thus 2 a b 1 ao> 2 a 2 < (a) ' that is 6 = 1' Moreover, we can shew that the diminution of the radius in passing from the equator to the pole will vary approximately as the square of the sine of the latitude. For we have that is therefore Hence if a r be small, we have approximately Thus a r varies as 1 ^ , that is, approximately as the square of the sine of the latitude. 57. The particular case in which the central force varies inversely as the square of the distance deserves to be noticed HUYGENS. 33 specially. In this case instead of equation (1) of Art. 55 we obtain -~.. -(2), where - 2 represents the central force. Thus the ratio of the axes is that of 1 to 1 4- -_ , and, taking -^ for aw 2 - - j , this ratio becomes that of 578 to 579, which is Oy Ob almost identical with that obtained in Art. 54. 58. If &) 2 a = \, the equation (1) of Art. 55 is equivalent to \f a? = 2ax, giving two parabolas, as Huygens observes. He seems in consequence to accept without hesitation, for relative equilibrium, a figure of revolution in which the two parts meet so as to produce an abrupt change of direction : see his page 157. 59. In his pages 159... 168 Huygens makes some interesting remarks on various points in Newton's Principia. Huygens does not admit that all particles of matter attract each other, but he does admit a resultant force exerted by the Sun or by a Planet, arid varying inversely as the square of the distance from the centre of the body. He states that he himself had not extended the action of pesanteur so far as from the Sun to the Planets, nor had he thought of the law of the inverse square : he fully recognises Newton's merits as to these points. 60. We must notice the value he obtained for the increase of gravity in proceeding from the equator to the pole. He adopts the ratio of the axes which he had found for the case of a constant force, and assumes that it will hold when the central force varies inversely as the square of the distance. Hence since the polar radius is -^ part shorter than the equatorial radius, gravity / o T. M. A. 3 HUYGENS. increases by ^-^ part in passing from the equator to the pole. And by reason of the absence of centrifugal force at the pole there is another increase of 7-, part, Thus on the whole there is an 2 increase of . He thinks that observation does not confirm this large increase. We know now that the increase is not so large as Huygens made it. His error arises from his assuming that the Earth's attraction is a single central force varying inversely as the square of the distance from the centre, instead of calculating the value of it from the form of the Earth. 61. Huygens expresses himself as much pleased with Newton's method of comparing the attraction at the surfaces of the Earth, the Sun, Jupiter and Mars: see his page 167. Huygens alludes to the very different estimates which had been made of the Sun's distance from the Earth : Newton took this to be 5000 times the Earth's diameter, Cassini to be 10000 times ; Huygens himself had taken it to be 12000 times. 62. Huygens also refers with pleasure to the researches of Newton respecting the motion of projectiles in a resisting medium. Huygens says he had himself formerly investigated this subject, assuming the resistance to vary as the velocity; after he bad finished his investigations he learned from the experiments made by the Academy of Sciences at Paris that the resistance in air and in water varied as the square of the velocity. He here gives the results of his original investigations without the demonstrations: see his pages 170... 172. It will furnish a good exercise for students to verify these results, which must have been obtained with some difficulty in the early days of the Integral Calculus. The results will be found to be all correct, except that on the middle of page 171 we ought to read terminal velocity instead of velocity with which the ground is reached. The phrase terminal velocity is due to Huygens; see his page 170. Huygens makes a few remarks on motion in a medium where the HUYGENS. 35 resistance varies as the square of the velocity ; but he considers only a particular case of vertical motion, and a particular case of oblique motion. The general problem, he truly says, is very difficult if not impossible. 63. Huygens finishes with a statement of properties of the exponential or logarithmic curve ; he does not give demonstrations, but they can be easily supplied. 64-. On the whole we may say that the chief contribution of Huygens to our subject is the important principle of fluid equi- librium, which we have noticed in Art. 53. He also first solved a problem in which the form of the surface of a fluid in relative equilibrium under a given force was accurately determined; see Art. 55. The result has become permanently connected with our history for a reason which we will now explain. The assumption that the attraction of the Earth varies inversely as the square of the distance from a fixed point is equivalent to the hypothesis that the density of the Earth is infinite towards the centre. This remark is in fact due to Clairaut ; see the Philoso- phical Transactions, Vol. XL. page 297. It is sometimes ascribed to Huygens himself; as in Barlow's Mathematical Dictionary, article Earth. But, as we have seen, Huygens preferred to consider the attractive force as constant ; and this is very different from the notion involved in Clairaut's remark. Laplace is not quite accurate in the Mecanique Celeste, Vol. v. page 5, where he omits all notice of the constant force, and says that Huygens supposed the force to vary inversely as the square of the distance from a point. 65. An important error has been sometimes made by repre- senting the researches of Huygens on the Figure of the Earth as preceding those of Newton in the order of time : for example, this is asserted in Barlow's article just cited. Svanberg also has com- pletely misrepresented the relative positions of Newton and Huy- gens : see his Exposition des operations faites en Lapponie. . .pages iii. . .v. The truth is that before the Addition to Huygens's Discourse the only remark on the. subject is the suggestion on page 152, that the Figure of the Earth is that of a sphere flattened at the poles ; 32 36 HUYGENS. and even this occurs in the part which treats on pendulums, written, as Huygens himself states, long after the greater part of the Discourse. The researches on the Figure of the Earth are really contained in the Addition, which as Huygens himself states was written after reading the Principia. There are two causes which might have led to this error in dates. In the first place, as Huygens was senior to Newton, it was natural in histories of science to give an account of the life and works of Huygens before those of Newton; this, for example, is. the course adopted by Bailly in his Histoire de I'Astronomie Moderne. Then a hasty glance at his Vol. in. page 9 might mislead an incautious reader. In the second place, it was natural to notice the partial and imperfect attempts of Huygens before proceeding to Newton's nearly complete solution; this, for example, is the course adopted by Clairaut in the Introduction to his Figure de la Terre. CHAPTER III. MISCELLANEOUS INVESTIGATIONS UP TO THE YEAR 1720. 66. THE present Chapter will contain an account of various miscellaneous investigations up to the year 1720. It is my design to write the history of the Theories of At- traction and of the Figure of the Earth ; and I have endeavoured to include all the memoirs and works which relate to these subjects. I do not profess to discuss the measurements of arcs and the observations of pendulums; but I shall briefly notice the more important of these operations in their proper places. 67. There are many writers to whom the student may be referred for accounts of the attempts made in ancient times, and in the early days of modern science, to ascertain the figure and dimensions of the Earth. Thus, for example, we may mention Cassini's De la Grandeur et de la Figure de la Terre, and Stay's Philosophies Recentioris Yol. n. with the notes by Boscovich. More recent works are the article by Professor Airy on the Figure of the Earth in the Encyclopaedia Metropolitana, and the article by the late T. Galloway on Trigonometrical Survey in the Penny Cyclopaedia. 68. Some interest attaches to the operations of Richard Norwood, which he has recorded in his Seaman's Practice, pub- lished in 1637. He says on his page 4: Upon the \\th of Jane, 1635, I made an Observation near the middle of the City of York, of the Meridian Altitude of the Sun, by 3S MISCKLLANKurS INVKSTICATIONS To 17 '20. an Arch of a >'//,,..,' i t' more than ~> Fo..t Semidiameter. and found the apparent Altitude of the San that \> \ at Nooll t" be .VJ deu r . '.''.'< lain. 1 had ids.) formerly upon the 11/7, of . ; . .-..,.!..,. ,o hi. 1 '."., observed in the City of A*/////*;,/, near the Tower, tlm apparent Meridian Alti- tude of the Sun. and {bund, the same to he (ii 1 ih-;. 1 min. And seeing the Sun's Declination upon the 1 1 //< ( l ; iy <>{' Jt'ic. 1 ().")">. and upon the \\tlt dav of .////^, hi.")."), was one and the same, witl out any sensible diilerence : and because these Altitudes diil'er but little, we .-hall Hot need to make any alteration or allowance, in respect of I >echnatien. Reflection, or .Parallax: \\heivfore subtracting the lesser apparent .Mtitude. namely .V,.i d-^-. .">." min. fr"in the greater (\'2 de^. 1 min. tin re remains '2 de^. L'S min. which i- the ditlerenee of Latitude of these two Cities. Ham --ly. of Lmnlnt and York. \\ will bo -eon that Norwood does not expressly say with what instrument ho observed tho Sun's altitude at London; ho lavs stivs- oil tie- f;'>'i that \\\" .ili.-i i-v; : '. : at L .ndon and at York wore made i n ii:< i same nay oi the in inlh. lie d.-! ermiiic.'d the di-i'iie- betw en 'i erl: and London in tho manner \\hieh he explain- on his p-'VJ' : ^ el ha\n i:; de ()l)ser\'a ion a; Y<>rL\ as aforesaid, I mea- sured foi- the ni' ' part ; the \\"ay from tin nee t > /.< , '/< : and where I measured n<t. I paced, (wherein thi'oii^li ('u.-tom ! us allv come \<T\' near the 'IVuth) observing all the wa\" a> I came with a Ciiviim- 't-i'enti'i 1 a.ll the principal .\ni:ids of Pi.-:;iion. or \\'indinLrs of the \Yav. (with eonvc.-nient allowance for other lesser \\'indimj.-. Ascents and I )e-e<-n! - .; : so that 1 mav atlirm the Ivxperiment to be near the Truth. Norwood made the distance between York and London f)H!l r-hains. each of !)|) fo,-t. lie deduced for the leii-lh oi' a doureo of tie- mui'idian .'JOni'i; (bet. Tlii- is inaivr to the ti'uih than III I'J 111 lia\ o boe]| e\peet od )V' '111 t he l'i ill- !| I'tOilo of 1 1 1 1 -a -- II IV H I O 1 1 1 i tie' modern result would lio soniowhat less than :ih")()(K> fed. I' li-'i- 1> 'ii -uppo-.-d that Norwood's woi'k bad boon for^ottoii b'-foro No\\tins tiino; but K'l-aiid i- st roiiid \- a^'MllisI tins sup- |)o.-,lt)oli: St.'O bl- //tuft/rim/ A'.s.sv/ //. . . p;e_;'o 1-. Ne\\"loli does Hot re for ill hi- 111 t i-dition to Norwood's \alue of a deirreo ; but he MISCELLANEOUS INVESTIGATIONS TO 1720. 39 docs in the second edition. Newton quotes the 367196 feet, which he says is 57300 Paris toises. The number of toises ob- tained will, of course, depend on the proportion of the English foot to the French foot. Cassini made the English foot to be | ; of the French foot ; see the De la Grandeur et de lor Figure de la Terre, pages 154, 251, and 282: this would give 57374 toises. Bailly in his Histoire de I 'Astronomic Moderne, Vol. II. page 342, gives 57442 toises, and draws attention in a note to Newton's smaller value. The comparison of English and French standards of length has, of course, been carried to minute accuracy in mo- dern times. See, for example, Airy, Figure of the Earth, page 217. G9. Richer made observations of the length of the seconds pendulum at Cayenne in 1672 : Varin, Des Hayes, and Du Glos made similar observations at Goree and at Guadaloupe in 1682. These observations are given in the Recueil d' Observations faites en Plusieurs Voyages.... Folio, Paris, 1693. Newton states the re- sults in the third edition of the Principia, omitting the name of Du Glos. It would seem from Newton's words that the same length was obtained at Martinique as at Guadaloupe: but the original account does not mention pendulum observations at Martinique. These observations had, however, been published before 1693 ; see Lalande's Bibliographic Astronomique, page 327 : thus they were accessible to Newton for his first edition, as we have men- tioned in Art. 36. Richer's observations are also given in Vol. vn. of the ancient Memoir es of the Paris Academy. 70. In Number 112 of the Philosophical Transactions, which is dated March 25, 1675, there is an account of Picard's survey of an arc of the meridian ; the Number forms part of Volume x. of the Transactions ; the account occupies pages 261... 272 of the volume ; it begins thus : A Breviate of Monsieur Picarts Account of the Measure of the Earth. This Account hath been printed about two years since, in French ; but very few Copies of it being come abroad, (for what reasons is hard to 40 MISCELLANEOUS INVESTIGATIONS TO 1720. divine;) it will be no wonder, that all this while we have been silent of it. Having at length met with an Extract thereof, and been often desired to impart it to the Curious ; we shall no longer resist those desires, but faithfully communicate in this Tract what we have re- ceived upon this Argument from a good hand. The account notices an attempt made by the Arabians to measure an arc of the meridian : a Station being chosen, and thence Troups of Horsemen let out, that went in a straight line, till one of them had raised a degree of Latitude, and the other had deprest it ; at the end of both their marches, they who raited it, counted 56| miles, and they who deprest it, reckon'd 56 miles just. This is not quite faithful to a description given by Picard, from which it may have been derived, which can be seen in Bailly's Histoire de V Astronomic Moderne, Vol. i. page 581. Picard does not mention Horsemen ; and he does not explicitly say which of the two parties obtained the longer measure. 71. In Number 126 of the Philosophical Transactions, which is dated June 20, 1676, and forms part of Volume XL, we have a notice of what Norwood effected. The following is the be- ginning of the notice : Advertisement concerning the Quantity of a Degree of a Great Circle, in English measures. Some while since an account was given concerning the Quantity of a Degree of a great Circle, according to the tenour of a printed French Discourse, entituled De la Mesure de la Terre. The Publisher not then knowing what had been done of that nature here in England, but having been since directed to the perusal of a Book, com- posed and published by that known Mathematician Richard Norwood in the year 1636, entituled 2'he Seaman s Practice, wherein, among other particulars, the compass of the Terraqueous Globe, and the Quan- tity of a Degree in English measures are deliver'd, approaching very near to that, which hath been lately observ'd in France ; he thought, it would much conduce to mutual confirmation, in a summary Narrative to take publick notice here of the method used by the said English Mathe- matician, and of the result of the same ; which, in short, is as follows : Tbe " Publisher" here means H. Oldenburg who was Secre- tary to the Royal Society. MISCELLANEOUS INVESTIGATIONS TO 1720. 41 An English translation of Picard's account of his survey of an arc of the meridian was published in 1687. The bulk of the volume in which it was included seems to have consisted of a translation of Memoirs on the Natural History of Animals. The Natural History was translated by Alexander Pitfield, and Picard's account by Eichard Waller. See Philosophical Trans- actions, Number 189, page 371. 72. A Discourse concerning Gravity, and its Properties, wherein the Descent of Heavy Bodies, and the Motion of Projects is briefly, but fully handled : Together with the Solution of a Problem of great Use in Gunnery. By E. Halley. This memoir is published in Number 179 of the Philosophical Transactions; the Number is for January and February, 1686, and forms part of Volume xvi. : the memoir occupies pages 3... 21 of the number. I notice this memoir for the sake of a fact to which Newton refers in the second edition of the Principia, Book ill. Prop. XX. Halley says : 'Tis true at S. Helena in the Latitude of 16 Degrees South, I found that the Pendulum of my Clock which vibrated seconds, needed to be made shorter than it had been in England by a very sensible space, (but which at that time I neglected to observe accurately) before it would keep time ; and since the like Observations has been made by the French Observers near the Equinoctial : Yet I dare not affirm that in mine it proceeded from any other Cause, than the great height of my place of Observation above the Surface of the Sea, whereby the Gravity being diminished, the length of the Pendulum vibrating seconds, is pro- portionably shortned. The " Problem of great use in Gunnery," which Halley solves, is one which we now enunciate thus: To determine the direction in which a body must be projected from a given point with a given velocity, so as to bit a given point. Halley considers his solution superior to those which had been previously given ; he says the problem was " first Solved by Mr Anderson, in his Book of the Genuine use and effects of the Gunn, Printed in the year 1674." 42 MISCELLANEOUS INVESTIGATIONS TO 1720. Halley observes, that for a given horizontal range the velocity is least when the angle of projection is 45. He says: This Rule may be of good use to all Bombardiers and Gunners, not only that they may use no more Powder than is necessary, to cast their Bombs into the place assigned, but that they may shoot with much more certainty, for that a small Error committed in the Elevation of the Piece, will produce no sensible difference in the fall of the Shot : For which Reasons the French Engineers in their late Sieges have used Morter-pieces inclined constantly to the Elevation of 45, proportioning their Charge of Powder according to the distance of the Object they intend to .strike on the Horizon. According to theory the horizontal ranges should be equal for two different angles of projection, one as much below 45 as the other is above 45; and Halley states that experiments shew there is little difference in the ranges, especially for large shot : see his page 20. 73, Thomas Burnet, master of the Charter-house, published towards the end of the seventeenth century his Sacred Theory of the Earth, first in Latin and afterwards in English. The work related to geology and the Mosaic cosmogony, and naturally gave rise to much controversy. I shall, however, not attempt to follow the details of this controversy, as it is but slightly connected with our subject; but content myself with noticing the contributions of one writer, Keill, whose name is not unknown in the history of mathematical science. The work of Keill now to be considered is entitled An Exami- nation of Dr Burners Theory of the Earth. Together with some remarks on Mr. Whiston's New Theory of the Earth. By Jo. Keill, A.M. Coll. Ball. Ox. 1698. The book contains 224 pages in octavo, besides the title-page and the dedication "to the Reverend Dr Mander, the worthy master of Balliol College in Oxford." 74. The part of the work which most concerns us is Chapter VI., Of the Figure of the Earth, which occupies pages 101... 143. Burnet maintained that the Earth was not oblate but oblong. Keill says on his page 107 : MISCELLANEOUS INVESTIGATIONS TO 1720. 43 I come now to examin the Theorists reasons by which he proves the Earth to be of an Oblong Spheroidical figure. He tells us that the fluid under the aequator being much more agitated than that which is towards the IVles which describes in its diurnal motions lesser arches, and because it cannot get quite off and fly away by reason of the Air which every way piv.ssrs upon it, it could no other wayes free it self than by flowing towards the sides, and consequently form the Earth into an Oval figure. Keill maintains, on tbe contrary, the oblateness of the Earth ; lie gives substantially the two investigations of the ratios of the axes which were then known, namely, that of Huygens, which assumed the resultant attraction to be constant, and that of Newton, which assumed the attraction between particles to vary inversely as the square of the distance. Keill also gives, after Huygens, a very clear account of the effect of centrifugal force on the position of a pendulum, and on the weight of a body. Keill does not refer to the work of Huygens, from which lie must have obtained a large part of his Chapter VI., namely, the Discours...de la Pesanteur ; but other works by Huygens are cited. 75. There is nothing new on our subject in Keill's work ; he merely reproduces what had been given by Newton and by Huygens. There are, however, some incidental mistakes which we should scarcely have expected from a distinguished member of a distinguished college. On his page 41 he says, " by calculation it will follow that a body would run down four thousand miles in the space of twenty- three seconds, abstracting from the resistance of the air." He must mean twenty-three minutes. On his page 1 50 he says, " for the ninty ninth power of 2 is a number which if written at length would consist of a hundred Figures." We know that 2" consists of 30 figures. On his page 156 he has an angle of which the tangent is - ; oU he makes the sine '19594, which is teo times too great: by cor- recting the error his own argument is much strengthened. 44 MISCELLANEOUS INVESTIGATIONS TO 1720. Oil his pages 160 and 161 he has some calculations, which he begins by stating that a perch is 10 feet, and which he continues on the supposition that a perch is 20 feet. 76. Keill's most serious mistake is one which it is very natural to make ; but, unfortunately, he is extremely incautious in drawing attention to it. He says on his pages 138 and 139 : Now tho' I have already determiued the Earths Figure from other Principles ; Yet to comply with the Theorist in this point, I will give him an account of a Book whose extract I have seen in the Ada Eru- ditorum Lipsice publicata for the year 1691. written by one Joh. Casp Eisenschmidt^ a German who calls himself Doctor of Philosophy and Physick. The Title of the Book is, Diatribe de Figura Telluris Elliptico- Sphceroide. And it is Printed at Strasburg in the year 1691.... Keill then proceeds to give some account of the book. Accord- ing to Eisenschmidt, the measurements hitherto made of the length of a degree of the meridian in various latitudes shewed that the length decreased as the latitude increased; granting this to be the case, Eisenschmidt inferred quite correctly that the Earth was of an oblong form. But Keill says on his pages 140 an(J 141 : None but a man of prodigious stupidity and carelessness could reason at this rate ! If he had asserted that the Earth was of an Oval Figure because Grass grows or Houses stand upon it, it had been something excusable ; for that Argument tho it did not infer the con- clusion, yet it could never have proved the contradictory to be true. But to bring an Argument which does evidently prove that the Earth has a Figure directly contrary to that which he would prove it has, is an intolerable and an unpardonable blunder. . . Keill's error consisted of course in misunderstanding what was meant by a degree of the meridian. Keill supposed that the difference of latitude of two places on the same meridian is the angle between straight lines drawn from these points to the centre of the Earth ; whereas in this subject, the difference of latitude means the angle between the vertical directions at the two places. MISCELLANEOUS INVESTIGATIONS TO 1720. 45 77. Keill's Chapter IV. is entitled, Of ike Perpendicular position of the Axis of the Earth to the plane of the Ecliptick. This Chapter contains some interesting matter ; though it is not con- nected with our subject. Bumet held that in the primitive Earth, the axis of the Earth's rotation was perpendicular to the plane of the ecliptic. Keill is thus led to consider the advantages resulting from the inclined position which we know the axis actually has. He infers by calculation, that places whose latitude exceeds 45 receive more heat from the Sun than they would do if the axis of the Earth's rotation were perpendicular to the ecliptic; while other places receive less heat. Keill derives his method, and some of his results, from a paper by Halley in the Philosophical Transactions, Number 203. On Keill's page 75, the first and second entries with respect to the Sun in Cancer ought to change places ; Halley is correct. 78. Keill on his page 70 charges Dr. Bentley with error for saying that "tho the axis had been perpendicular, yet take the whole year about we should have had the same measure of heat we have now." But it is obvious that Bentley is right in a certain sense ; namely, that the whole heat received by the Earth is the same in the two cases. I am sorry to see that Keill goes on to shew that Bentley is to be numbered among the advo- cates of an error which has at all times been popular; accord- ing to Keill, page 70, ...in the same Lecture, he confidently saies, that 'tis matter of fact and experience that the Moon alwaies shews the same Face to us, not once wheeling about her own Centre, whereas 'tis evident to any one who thinks, that the Moon shews the same face to us for this very reason, because she does turn once, in the time of her period, about her own Centre. The Lecture to which Keill alludes is the " last Lecture for the Confutation of Atheism" 79. Keill published another work on the same subject as the former ; it is entitled A n Examination of the Reflections on the Theory of the Earth. Together with a Defence of the Remarks on Mr Whistons New Theory. The book contains 208 pages in 4(i MISCKLLAXK.ors IXYKSTKJATK >NS TO 17-0. octavo, beside- tlie title-jKiLiv. It furnishes nothing eonneeted with our subject except another reference to l)r Kismscliiiiitlt. Kedl seems determined to remain uncoiivinced of Ins error; lie savs on liis Jt;;i4'e I 00 : Our D'f'ii'l'-r tells us, that Dr. /;'/>'/ /.-</,//;, / supposes tin- Vertical Lines or Lines of Oruvity. to lie drawn at ri^ht Angles to the Tangent of each respective Ilori/on. What Dr. Ei*< i uxt'hni\<lt does really supjtose I know not, hut 1 am .-lire he cannot sujtjK.se a tiling more alt.-'iird than what our Author makes him suj>j>ose in this ji]ace. For that the Line of direction of heavy l>odies is at ri^ht Angles with the Tancri-nt of the Ilori/ou, is to me sneli an inconijtrehensible sujijxisition, that I shall excu.-e mv self i'nnn considering of it, till the 1), /', ,!< -/ (v/ho 1 sujtpose \vould have us think he understands it) is at leisure to explain it. Keill was suUseijUcntly artpointed Savilimi IVofe.-sor of As- tronomy at Oxford: k-T us lioju. 1 tli;it befoi tli;it time lie under- stood this siinjtle matter Avliieli had porp'exrd liim. (SO. In the Paris Mt'niotre* for 170*. [)ul)I5slied in !7o:>, \vc have two articles bearinir on our sultiect : l.oili uerur in the On pnn'os 1 11. ..1 Ki. 1li. -r- is a notice of sonu' olisei'val ions of the L-ii'.itli o!' tlie ^-or-oiids ]:eiid!ihuii i,:;.de by Oonp'ei iii lo!)7 at Lisbon, and in ICiJS at I^irnyl.M in liraxil. On ji;cj;e- ~\ *2(} . . .'\ '2-\ , there is a brief aceonnt of tin- operations nj> to tlie current date foiinc'cted v.'itb the 1 Freneh arc of the meridian. 81. In the Paris MSinnircx lor 1701, jmblisbe.l in 1 70 1, we have on pa-'e 111 of llie lii>tori-;d jjoi-tioii of llic vnlunif some ii'-ndulnn! observations made b\" ( )e- Haves in l(J!l!) and 1700: Xewlon stati > (lie I'l-sults in ibe third edition of the Priin-t^'nt. Ill tlie -Mine ViOllJie. there is \\ l!|fli:oi|- elltiiled I ><' frl J/J'- '//'(!,'( i ! i/'' /'< il/.-'f //</!, ,/',<> /,'i'i/nl jn-iJi,,'i<c /ttsijti a tr.r' I'lfnin'c^. ]>,- M. <',: ; ;. T;, , . ' ; ..:. , !;:). ..LSJ of the MISCELLANEOUS INVESTIGATIONS TO 1720. 47 operations in France. The substance of the memoir is reproduced in the first eighteen pages of the work De la Grandeur et de la Figure de la Terre. There is an account of the memoir in pages 96 and 97 of the historical portion of the volume. Here we have the error which Keill adopted, as we saw in Art. 76 : Mais en supposant, comrae il est fort vraisemblable, que cette diminution de la valeur terrestre d'uii degre, continue toujours de 1'Equa- teur vers le Pole, et en conservant d'ailleurs les hypotheses communes, on voit d'abord qu'un Meridien doit etre plus petit que 1'Equateur, et par consequent que la Terre est un Globe aplati vers les Poles. The passage was changed in another edition : see La Lande's . Astronomic, third edition, Vol. m. page 24. 82. In the Paris Memoires for 1702, published in 1704, we have a memoir entitled Reflexions sur la mesure de la Terre, rapportee par Snellius dans son Livre intitule, Eratosthenes Batavus. Par M. Cassini le Jtls. The memoir occupies pages 61... 66 of the volume : see also page 82 of the historical portion of the volume. Cassini shews that Snell's result was quite unsatis- factory. The memoir is substantially reproduced with additions in pages 287... 296 of the work De la Grandeur et de la Figure de la Terre. 83. In the Paris Memoires for 1703, published in 1705, we have a memoir entitled fiemarques sur les Inegalites du Mouvement des Horloges a Pendule. Par M. De La Hire. The memoir occupies pages 285... 299 of the volume : there is an account of it on pages 130... 135 of the historical portion of the volume. 84. David Gregory, Savilian Professor of Astronomy at Oxford, published there in 1702 his Astronomice Pliysicce et Geometricce Elementa: it is a folio volume containing 494 pages, besides the Title, Dedication, Preface, and Index. The work was reprinted in two quarto volumes at Geneva in 1726, with some additions by an editor who signs himself C. Huart, M. and P. S. A section of the work is devoted to the Figure of the Sun and the Planets: this section occurs on pages 268... 272 of the origi- nal edition, and on pages 408... 414 of the reprint. 48 MISCELLANEOUS INVESTIGATIONS TO 1720. David Gregory contributes nothing new to our subject. He repeats two mistakes from Newton, with rather increased emphasis. One mistake is the assertion that gravity at the surface varies inversely as the radius, instead of gravity resolved along the radius : see Art. 33. The other mistake is the assertion that if instead of being homogeneous, the central portion is denser than the rest, then the ellipticity is increased : see Art. 30, and Clairaut's Figure de la Terre, page 254. On the hypotheses that the figure is an oblatum, and that gravity varies inversely as the radius ; David Gregory gives a good geometrical demonstration of the theorem, that the increase of gravity in proceeding from the equator to the pole varies as the square of the sine of the latitude. On page 37 of his own edition, David Gregory stated the oblateness of the Earth as a fact. This is the only point at which the editor of the reprint ventures to correct the original author ; and on page 51 of the reprint we have this unfortunate note : Constat ex celeberrimomm Geometrarum observationibus, experi- xnentis et argumentis, Terrain quidem Sphseroidem esse, sed oblongain non verb depvessam versus Polos, contra quod affirmat Autor noster. Verum circa hanc qusestionein consulantur Historia et Commentarii Regise Scientiarum Academise anni praesertim 1720. Keill very naturally praised the work of his predecessor in the Savilian chair ; though with some extravagance of language. The following words occur in the Eicerche sopra diver si punti...of Gregory Fontana, Pavia, 1793, pages 93 and 94 : II famoso David Gregori nella sua elegantissima opera intitolata Astronomice Physicce et Geometricce jElementa, che dal celebre Giovanni Keil nella Prefazione della sua Introduzione alia vera Fisica ed Astro- nomia viene caratterizzata col pomposo elogio di opus cum sole et luna duraturum 85. A memoir by Keill is given on pages 97... 110 of Number 315 of the Philosophical Transactions. The Number is for the months of May and June, 1708 ; it forms part of Volume xxvi. which is for the years 1708 and 1709, and is dated 1710. MISCELLANEOUS INVESTIGATIONS TO 1720. 49 The memoir is entitled Joannis Keill ex ^Ede Christi Oxon. A.M. Epistola ad Cl. virum Gulielmum Cockburn, Medicince Doc- tor em. In qua Leges Attractionis aliaque Phy sices Principia traduntur. The memoir is reprinted at the end of the edition of Keill's Introductions ad veram Physicam... published at Leyden in 1739. 86. The memoir consists of thirty theorems ; many of them are merely enunciated ; others are supported by a short com- mentary. They are but little connected with our subject, being experi- mental rather than mathematical, and bearing on what we should call molecular attraction. 87. Keill speaks of Newton as Vir ingenio pene supra humanam sortem admirabili, dignusque cujus fama per omnes terras pervagata, coeli quos descripsit meatibus permaneat coseva. The immensity of space and of time with which Astronomy is concerned may cause but can scarcely justify the exorbitant language in which the achievements of those who cultivate the science are sometimes described. The expressions of Keill with respect to Newton may be compared with those which Arago uses when noticing Poisson's famous memoir on the permanence of the solar system : II aura etabli qu'a ce point de vue, le seul dont Newton et Euler se fussent preoccupe"s, les ge*ometres, ses successeurs, liront encore son beau Memoire dans plusieurs millions d'anne*es. CEuvres completes de Francois Arago, vol. n. page 654. S'il en e"tait besoin, le magninque Me"moire sur I'lnvariabilite" des grands axes, prouverait que Poisson avait un interest personnel a porter fc es regards, ses pense"es, sur des siecles si e"loigne*s. The same volume, page 696. 88. Keill says that he had thought about applying a prin- ciple similar to Newton's attraction for the explanation of terres- trial phenomena ; and had tested the notion by experiments. He adds : T. M. A. 4 UNIVERSITY OF CALIFORNIA 50 MISCELLANEOUS INVESTIGATIONS TO 1720. Meaque hac de re coiritata, abliinc quinqucnnio, Domino Xewtono imlicavi ; ex eo autem intollexi, cadem fere, qua 1 ipsc investigaveram, sibi diu ante animadversa fuisse. <SO. Almost tlif only passage in tin- memoir which directly concerns us presents a difficulty. Keill's Tlieorem XV. asserts tliat the attractive forces of perfectly solid particles depend much on their li^ures. He proceeds thus: Xam si jarva aliqua materise particula in laminam circnlarem inde- finite exigua3 crassitudiiiis formetur, et corjmsculum in recta per cen- trum transeunte et ad planuni circuli Xormali locctur ; sitque distantia corpusculi {equal is decimse parti semidiametri cireuli : vis cpia urgetur corpusculum tricesies minor erit, (juam si materia attrahens coalesceret in Sphseram, et virtus tutius particula? ex uno quasi jauicto l^hv.sico diffunderetur. Let J/ denote the mass of the particle, c the distance from the centre of the lamina of the attracted corpuscle, b the radius of the lamina. Then by the ordinary formula -we have the attraction 2J/ In the case of the sphere the attraction = . The ratio of the former to the latter is < } Since b = lOc, this ratio = 1 1 / -i- j I presume tricesies is intended for thirty times, though it is not contained iu the dictionaries. Ileiu-o Keill has --- instead of. oO oo T)io forrnulrr- for the attraction of a circular lamina and of a sphere arc implicitly given by JS'cwton ; so that there is no reason for the error. MISCELLANEOUS INVESTIGATIONS TO 1720. 51 90. The Paris Memoires for 1708, published in 1709, contain observations of the length of the seconds pendulum made by Feuillee in 1704 at Porto Bello and somewhat later at Martinique : see pages 8 and 16 of the volume. The anomalous results ob- tained were noticed by Newton in the second and third editions of the Principia. 91. We have next to advert to a paper published in pages 330. ,,342 of Number 331 of the Philosophical Transactions, which is for the months of July, August, and September, 1711. The number forms part of Volume xxvil. which is for the years 1710. ..1712, and is dated 1712. The title of the paper is Johannis Freind, M.D. Oxon. Prcelec- tionum Chymicarum Vindicice, in quibus Objectiones, in Actis Lipsiensibus Anno 1710. Mense Septembri, contra Vim materim Attractricem allatce, diluuntur. The paper is not mathematical. Freind had published a work on Chemistry, and the editors of the Leipsic Acta found fault with the use he made of the principle of Attraction. In this paper Freind maintains the truth and the importance of the principle. 92. In the Paris Mtmoires for 1713, published in 1716, there is a memoir entitled De la Figure de la Terre. Par M. Cassini. The memoir occupies pages 188... 200 of the volume. The arc of the meridian measured from Paris to the south of France, compared with the arc measured northwards, seemed to indicate that the length of a degree of the meridian decreased from the equator to the pole. This result suggested that the Earth is an oblongum. Accordingly Cassini so considers it ; and assuming that the excentricity of the generating ellipse is about jY he calculates a table of the length of a degree of the meridian for every degree of latitude. The memoir is substantially repro- duced in pages 237. ..245 of the work De la Grandeur et de la Figure de la Terre; but the table is there calculated for the excentricity ^. 42 52 MISCELLANEOUS INVESTIGATIONS TO 1720. Some introductory matter given in the memoir is not repro- duced in the work just cited. This matter contains short accounts of the opinions of Newton and of Huygens in favour of the oblate form of the earth. Then a contrary opinion is noticed at greater length, beginning thus : Tout au contraire, M. Einsenschmid celebre Mathematicien de Strasbourg... We have already learned the nature of this opinion : see Art. 76. There is an account of the memoir on pages 62... 66 of the historical portion of the volume. It is there remarked that sup- posing the length of a degree of the meridian to decrease from the equator to the pole, it would not follow, as had been erroneously suggested in the historical portion of the Memoir es for 1701, that the Earth is flattened at the poles: see Art. 81. 93. James Hermann published at Amsterdam in 1716 a quarto volume, entitled Phoronomia, sive de Viribus et Motibus corporum et fluidorum libri duo. We are concerned only with pages 361... 371 of the work. 94. Hermann solves Huygens's problem of the relative equi- librium of rotating fluid under the action of a constant force directed to a point on the axis of rotation. Hermann gives two solutions; one on Newton's principle of columns balancing at the centre, the other on Huygens's principle of the plumb-line. 95. Hermann also solves by both principles the problem in which the central force, instead of being constant, varies as the distance; in this case he shews that the figure is an oblatum. This is the first appearance of the problem and its solution. For the case of the Earth the ratio of the axes would be nearly as V288 is to V 2 89, that is, approximately as 577 is to 578. Hermann's investigations of both problems are correct and Fatisfactory. There is, however, a curious circumstance connected with his second problem. He notices that the result differs very much from that which Newton had obtained for the ratio of the axes of the Earth ; he does not expressly say that Newton was wrong, but he seems to imply that his own was the correct MISCELLANEOUS INVESTIGATIONS TO 1720. 53 result. He observes that neither Newton nor David Gregory had determined what the figure must be for equilibrium ; and this is certainly true. See, however, Boscovich, De Litteraria Expedi- tions, pages 4 42... 446. 96. In Newton's fluid mass, assumed to be an oblatum, so long as we keep to the same radius vector, the attraction varies as the distance from the centre, and so also does the gravity. And at the surface the gravity resolved along the radiiis vector varies in- versely as the length of the radius vector. Now Hermann notices these results ; though he seems to pay no attention to the limiting clauses which I have printed in Italics. Both results hold for Hermann's own fluid mass. Moreover, Hermann demonstrates a proposition which we may enunciate thus : Suppose a fluid mass in relative equilibrium under a centrifugal force and a central force to some point of the axis of rotation ; then if at the sur- face the gravity resolved along the radius vector varies inversely as the length of the radius vector, the attraction at the surface varies as the distance from the centre. Perhaps, from seeing that his fluid mass and Newton's had similar properties, Hermann inferred that Newton's figure and his own ought to be identical. But it is sufficient to observe that Newton's problem and Hermann's are essentially different. New- ton does not assume attraction to a fixed centre varying as the distance; he assumes that every particle attracts every other according to the law of the inverse square of the distance. It should have been a caution to Hermann that his own problem and Huygens's led to approximately the same result for the ratio of the axes, though the laws of force were very different ; thus from partial agreement he ought not to have expected universal agreement. 97. Hermann seems to have been much surprised at the proposition which, as we have said in the preceding Article, he demonstrates. He observes on his page 369: Hac verb proprietate posita, quod scilicet solicitationes gravitatis acceleratrices...tlistantiis a centre... reciproce proportionales sunt, quis 54 MISCELLANEOUS INVESTUiATlONS TO 1720. crediderit gravitates alsolutas eorporuin in iisdem punctis...eorum dis- tantns. . .directe proportionales esse ? Boseovieh, nearly fortv years later, expressed his surprise at ili'" 1 same result: see his De Littcrana Expeditions page 403, where lie savs : gravitates residua, 1 erunt accurate in superlicie ejus solidi in ratione reciproca distaiitiarum a centre, quod sane miruui videri possit, cum gravitates priinitlvse ibidem sint in rations directa distantiarum t'cirundem. It will be observed that what 1 call attraction Hermann calls iji'iicitas (tbs'tl nt(.t , and Boscovieh (jravittis prim i licet ; Avhat I call tjn.tt'itu Hermann calls .solititatio (ii-itcifMtift acceleratrLc,&iid Bosco- vich (/ru.cit'tx i'C'tfid'K.1. See Art. -."). 1'S. On his pap? -*>7-, ilermann discusses a pr<l)lem about i''.>tating fluid, which dees not concern our subject. Hero he {'alls into an error, which was pointed out by Clairaut in page 55 of his Piijure de l<t r l\'rre. fM). In the Paris Mcnwires for 171S. published in 1719, we have a memoir entitled DC la. Grandeur dc In l\ j rre ct de sa y'cj'ire. pur },[. (''j.^ini. It occupies pap-s IS0...1[)(5 of the volume; th're is an account of it on pages 04.. .00 of the historical portion (.if the' volume. Tin' ni'-nioir contains a notice of the labours of the ancients on the subject, and of the reeent operations in France. It is sub- stantially reproduced in the work J)e la Grandeur ct de la Fiyure dc la Terre, pages 1^...18 and 1 S<). . . ![)<>. 100. A\ e have no\v to eonsKh.'r the account of the measure- ihent of an are of the meridian tliroiigh France, which is con- t- iii ( ' ( l in the \\ork J ' } ( > Id (',r<ind<ar ct dc la l'"t<jurc de la Tcrre ; the work has also the title finite des Mr'nit/irex de I* A endemic liuytle des Sciences, .lnn< f e 171S. The date of publication is 1720. Tin- volume is in <|iiarto. It contains Title. Half-title and I able i>t ('ontcnN in (j pa _'>, an<l oOo' paL;< s <f text. There is MISCELLANEOUS INVESTIGATIONS TO 1720. 55 a small map of France, and 4 large maps shewing the meridian line of ParivS traced through the kingdom; there are also 15 plates. A list of the misprints in the work is given in the Paris Memoires for 1732, pages 512 and 513. 101. The volume is divided into two parts; in the first part the operations are described which relate to the arc extending from Paris southwards to the Pyrenees, and in the second part the operations are described which relate to the arc extending from Paris northwards to Dunkirk. The author's name is not given explicitly ; but we learn incidentally that it was J. Cassini : see pages 5, 10, 193, 302, 303, 304, 305. The operations which the volume records are the most accu- rate and important which had as yet been performed in connection with the Figure of the Earth ; and the account given of them is interesting and satisfactory. The instruments and the methods of using them are fully and clearly described, and the calculations exhibited in such a manner that they can be easily tested. 102. The determination of an arc of the meridian we are now considering is a continuation of the work commenced by Picard in 1669. Picard measured a base of 5663 toises near Paris ; then by a series of triangles he found the distance between the paral- lels of Malvoisine and Amiens to be 78850 toises, corresponding to a difference of 1 22' 55" in latitude: hence he adopted 57060 as the number of toises in a degree. See pages 273, 256, 281. It was afterwards proposed to extend Picard's' arc through France ; and the work was committed to D. Cassini and others : but it was interrupted in 1683. The work was resumed by D. Cassini, J. Cassini, and others in 1700, and the arc was ex- tended southwards to the Pyrenees. In 1718 the extension of the arc northwards to Dunkirk was commenced. See pages 4, 5, 191. In this extension many of Picard's triangles were em- ployed : see pages 191, 255. 103. All the triangles were calculated in succession from Picard's original base, which was not re-measured. Two bases of verification were measured, one near the Pyrenees, and the other 56 MISCELLANEOUS INVESTIGATIONS TO 1720. near Dunkirk. The difference between the measured and the calculated length was three toises in the former case ; but this was reduced by some necessary corrections of the angles : the difference in the latter case was about a toise. See pages 104 and 221. Both bases of verification were measured by wooden rods. In the former case four rods each of two toises in length were joined together, two and two, so as to make two rods each of four toises in length ; in the latter case three rods each of three toises in length were used : the lengths of the wooden rods were determined in both cases by the aid of the same iron rule, four feet long. See pages 99 and 219. Picard's original base had been measured by four rods each of two toises in length, which were joined together two and two, so as to make two rods each of four toises in length. See page 255. 104. The general result obtained is the following: from the southern arc which extended over nearly 6 19', the length of a degree was found to be 57097 toises ; from the northern arc which extended over rather more than 2 12', the length of a degree was found to be 56960 toises. This was considered to make it suffici- ently evident that the length of a degree of the meridian must diminish from the equator to the pole. Assuming then that the earth is an oblongum, the ellipticity is found to be - . See yo pages 148, 237, 243. A table is given of the length of a degree of the meridian in different latitudes on the Cassinian hypothesis : see Arts. 39 and 92. It is now well known that the length of a degree of the meridian increases from the equator to the pole ; the contrary opinion however, maintained by J. Cassini, found advocates for some years after the publication of the work we are now con- sidering. As we shall see, the erroneous determination deduced from the French arc was finally corrected by fresh operations. 105. Pages 255... 287 of the volume are devoted to the subject of Picard's measure of the Earth. As Picard's book was scarce, large extracts are given from it ; a few remarks are made which do not substantially affect Picard's accuracy. MISCELLANEOUS INVESTIGATIONS TO 1720. 57 Pages 287... 306 of the volume are devoted to the measure by Snell and the measure by Riccioli ; the value of both is quite demolished: see Art. 82. 106. A few remarks may be made on some incidental points. I offer with hesitation an opinion as to instruments ; but from the descriptions given it seerns to me very unlikely that either the geodetical or the astronomical angles could have been observed accurately to seconds as is professed. The astronomical instruments used at the north and south extremities of the arc were different; the former had an error of 3 seconds in a degree from false centering. See pages 142, 223, 233. On pages 225... 230 we have an account and an explanation of a fact stated to be then observed for the first time, which gave much trouble until it was understood. The fact is this in modern language : any star which is not an equatorial star does not strictly run along the horizontal wire of a transit instrument as it crosses the meridian of the observer ; thus in determining the zenith distance from observations of the star when it is not accu- rately on the meridian, it is necessary to allow for the curvature of the path. Speaking of the distinction of the regions of the Earth into East, West, North and South, our author gives a paragraph which I quote for the sake of its last example ; see his pages 20, 21. Cette meme distinction des regions fut observSe dans la construction du Temple de Jerusalem. Nous voyons aussi qu'elle a ete" iinitee dans la construction des premiers Temples Chretiens, quand on 1'a pu faire commodement, et meme dans la situation de la Maison de Notre-Dame de Lorette, comme nous 1'avons observe nous-memes apres plusieurs autres Mathematiciens. Much importance was attached to the precaution of taking the observations of stars at the same season of the year : see pages 144 and 231. It seems to have been made out even then that the altitudes of the stars varied at different seasons. We know now that the Aberration of Light would certainly cause such variations. 58 MISCELLANEOUS INVESTIGATIONS TO 17-0. Speaking uf tlie largest Egyptian pyramid our author says on Li- page 1 .">4 : 11 y a lieu de .sY't. timer. qne M. Graves Mathematician Anglois, dans sa Pyraiiiidograpliie, ait tnmve la kisi> do cettc Pyramids, me.suree par les Triangles, de <>''.") pied* de Londros. . . . The error is certainly large ; for according to trustworthy statements the base was originally 704- feet, and is now 74(5 feet: see HerschcTs Familiar Lectures on Scientific subjects, page 4i?7. The inaccurate measurer was John Greaves, Savilian Professor of Astronomy at Oxford. 107. AYe may state here, though a little nut of chronological order, that a German translation of the De Id Grandeur et de hi Fi'i'ire de la Terre wa> publislied in 17-M at Arnstadt and Leipzig. This is entitled Mathem at incite nnd fjc/Ki>ie Abhandluny von der Fi(jv.r and G rosse der Krden, r i.hero is a preface 1>\ J. A. Klimmen, from ^hieh we learn that the translator, who>e name is not stated, did not live beyond the commencement of the printing. The translation is in a small octavo form; there are no maps, but the other plates of the original are c-"pied, >n a diminished scale. The misprints pointed out in the Paris Manuires for 1732 are corrected. It seems strange that a translation should have been published when the original work was just about to be superseded. In 17'W astronomical observations had been made by Maupertuis, Clairant, Camus and Le Monnier, in order to determine anew tin 1 length of a degree between Paris and Amiens; and in 17-1-0 Picard's base was remeasured: in 1744 the work entitled L<i Meridienne de Paris 1 OS. An account of the work Ih>. l<i Grandeur et de l<t Figure do In Terre is given in the, Pai'is Mi'moirca for 17-1, ])ublished in 17-'i. r ni- account is on pages (('>.. .77 <f the Listoric-al portion of the volume: it fnrni.-hes n-fcn-nces to preeeding volumes of the M'-nntirt'.-i in \shich tin.- .-ul)ject had been noticed. There is nothing of importance in tli*- aocount. MISCELLANEOUS INVESTIGATIONS TO 1720. 59 The following sentence, so far as it is intelligible, suggests a proceeding which may very naturally have been adopted; but I do not know what authority there is for the statement. En tirant d'un Lieu line perpend icu'aire sur la Meridienne, pour avoir la distance e ce Lieu par rapport a elle, on a considere s'il en etoit proclie, ou s'il ne 1'etoit pas. Dans le premier cas la perpendiculaire etoit la distance asses juste, mais dans le second, cette perpendiculaire representoit un petit arc de Cercle, et Ton avoit e"gard a la difference de 1'arc et de la Corde, qui etoit la distance cherchee. Page 146 of the work seems to approach nearest to the latter part of the above statement. 109. We have now to consider a memoir by Mairan, entitled Recherclies Gfometriques sur la diminution des Degres terrestres, en allant de I'Equateur vers les Poles : Ou I'on examine les consequences qui en resultent, iant a I'egard de la figure de la Terre, que de la pesanteur des corps, et de I 'accourcissement du Pendule. This is contained in the Paris Memoir es for 1720, published in 1722. The memoir occupies pages 231... 277 of the volume. The memoir may be described generally as consisting of mis- applied mathematics. Mairan was a Cartesian and a Cassinian ; so that he upheld the system of vortices, and the oblong form of the Earth. There is an account of the memoir in pages 65... 79 of the historical portion of the volume ; this is I presume by Fontenelle, who was then Secretary of the Paris Academy of Sciences : Mairan's opinions seem here to be accepted without hesitation. 110. Mairan shews that if the length of a degree of the meridian decreases from the equator to the pole, the polar dia- meter must be the longest. He compares the effect produced by centrifugal force at a place in the same latitude on the surface of a sphere, an oblong body, and an oblate body ; the latitude being determined in each case by the angle between the normal to the surface and the plane of the equator. Part of his page 244 is unsatisfactory, but it can be easily corrected. 60 MISCELLANEOUS INVESTIGATIONS TO 1720. 111. Mairan supposes that the Earth was originally of an elongated form, and that the amount of elongation was diminished by the centrifugal force, but not entirely destroyed. See Bailly's Histoire de I' Astronomie Moderne, Vol. n., page 641. Mairan's Proposition VIII. on page 253 is a striking example of the vagueness of the mechanical language of the period.. He speaks about the centre of the Earth sustaining a part of the effort of gravity: it is difficult to attach any meaning to such an ex- pression. 112. Mairan has a long discussion on the direction of gravity at different points of the interior of the Earth. Suppose that through any point of the interior a surface is drawn, similar, similarly situated, and concentric with the external surface ; Mai- ran takes the normal to this surface at the point for the direction of gravity. Then, to determine the lines of direction of gravity, he solves what we call a problem of orthogonal trajectories ; the curves which are cut at right angles being ellipses, similar, similarly situated, and concentric. Thus his result coincides with what we should obtain in seeking the lines of force inside a homogeneous mass of rotating fluid, supposing it in relative equi- librium. Mairan seems to attach great importance to the matter ; he thinks his lines of direction may extend beyond the Earth to the boundary of the terrestrial vortex ; he admits however that there is little prospect of verifying his result by observation : see his page 263. 113. But the most extraordinary part of the memoir is that which treats of the variation of gravity at the surface of the Earth. Newtonians and Cassinians agreed in admitting, as a result of ob- servation, the diminution of gravity in passing from the pole to the equator. Huygens's notion that the resultant attraction is constant at all distances from the Earth's centre would not reconcile this fact with an oblong form of the Earth. Newton's law of attraction according to the inverse square of the distance directly contra- dicted the oblong form. Accordingly, Mairan had to invent a law ; he suggests and rejects various other absurdities before he produces that which he adopts : we will describe this in modern language. MISCELLANEOUS INVESTIGATIONS TO 1720. 1 Mairan holds that at every point of the surface of a body of revolution the force of attraction would vary inversely as the product of the two principal radii of curvature at the point. His reason for this assumption depends on the fact that adjacent normals to the surface, taken in the plane of the meridian, inter- sect at one centre of curvature, while adjacent normals to the surface, taken in the plane at right angles to the meridian, intersect at the other centre of curvature. With this arbitrary law, Mairan triumphantly shews that the oblong form makes gravity decrease from the pole to the equator, which agrees with observation; while the oblate form makes gravity increase from the pole to the equator. He prudently abstains from numerical calculation which would test the extent of his agreement with observation. If we take an oblongum, we find that Mairan's law makes the attraction at the pole bear to the attraction at the equator the ratio of the fourth power of the polar diameter to the fourth power of the equatorial diameter; thus, assuming with J. Cassini and Mairan, the ellipticity to be about , the diminution of gravity in passing from the pole to the i/O equator would be about -^ of the gravity at the pole, besides that caused by the centrifugal force : this is extravagantly greater than observation suggested. It would be difficult to find a more striking example of mis- placed ingenuity than the pages 264... 276 of the memoir, which are devoted to Mairan's arbitrary law. 114. With respect to the equation which Huygens obtained, as we stated in Art. 55, Mairan says on his page 253 : M. Huguens a donne PEquation algebrique de la courbe ge*ne"ratrice du sph Oroide applati, par rapport a la Terre supposee primitivement spheVique ; et M. Hermann, qui avoit trouve la meme courbe par le calcul integral, dans sa re"ponse a M. Nieuwentiit, Pa encore donne"e par synthe"se, et avec la construction, dans sa Phoronomie. I have not seen the first production of Hermann, to which Mairan refers: I have noticed the second in Arts. 93... 8. (52 MISC KLLANKOUS I N V KSTIv ; ATIoNS TO 1720. 115. The writers who have appeared before us in the present Chapter added nothing to Newton's investigations on Attraction .and on tin- Figure of the Karth ; while under the powerful in- fluence of ]). I'assini and ,1. Cassini doubts had arisen as to the real shape of the Karth. But the true tlu-orv ultimately < r ained I . * * O the support of decisive researches and measurements. The next three Chapters will he devoted to three eminent mathematicians who all contributed essentially to the advance- ment of our subject. Maupertuis adopted and explained Xewton's propositions on Attraction and on the Figure of the Earth-; and he conducted an expedition to Lapland, for the measurement of an arc of the meridian, the result of which was fatal to the Cassinian hypothesis. James StirlH'ig enunciated without demon- stration approximate propositions respecting the magnitude and the direction of the attraction of a homogeneous oblatum at its surface; and he implicitly established Xewton's postulate: see Art. 4-4-. Clairaut produced several valuable memoirs ; in particular, during his stay in Lapland, lie found leisure to compose one on the same subject as Stirling's : another memoir led the way to the investigations of the Figure of the Earth, supposed hetero- geneous. These two memoirs were subsequently embodied by Clairaut in a work of enduring interest and importance. CHAPTER IV. MAUPERTUIS. 116. WE shall notice in this Chapter the various memoirs which Maupertuis contributed to our subject. 117. A memoir is given on pages 240... 256 of Number 422 of the Philosophical Transactions. The Number is for the months of January, February, and March, 1 732 ; it forms part of Volume xxxvil. which is for the years 1731 and 1732, and is dated 1733. The memoir is entitled De Figuris quas Fluida rotata induere possunt, Problemata duo ; cum conjectura de Stellis quce aliquando prodeunt vel deficiunt; et de Annulo Saturni. Authore Petro Ludovico De Maupertuis, Regice Societatis Londinensis, et Aca- demice Scientiarum Parisiensis Socio. 118. In the first problem, fluid is supposed to rotate with uniform angular velocity round a fixed axis, and to be attracted to a fixed point in the axis by a force which varies as any power of the distance. Maupertuis uses Newton's principle of balancing columns, and investigates the equation which determines the form of the surface for relative equilibrium. He restricts himself, as we should say, to space of two dimensions ; but a modern reader will have no difficulty in solving the problem generally, and the result will coincide with that of Maupertuis. 119. The second problem is enunciated thus : Posito quod materia fluens circa axern extra fluentum sumtum, attrahatur versus centrum in hoc axe positum vi alicui distantiae a Oi MAUPKUTUIS. centre dignitati proportionali ; clum iuterea propter fluenti partium attractionem nnituani, sit altera attractio versus aliud centruiu intra fluentura sinntuin, qiue in quavis sectione fluent! revohitionis perpen- diculariter per centrum exterius facta, sit alicui distantia* a centro interior! dignitati proportionalis : invenire iiguram quam flucntuin induct. In the solution of this problem also, Maiipertuis restricts himself to space of two dimensions ; but it may be shewn by a more general process that his result is correct. Take the axis of z for that of rotation; let a) be the angular velocity ; and (>, y, z) any point of the fluid. Then in the usual way, we may suppose the system reduced to rest, if we impress forces u?x and ory parallel to the axes of x and y respectively. Let there be a force directed to the origin, denoted by \r m , where r = \/(or-f y~ + z"}. Besides this there is to be a force of a certain kind, arising from the attraction of the mass itself. This mass is supposed to form a symmetrical ring-shaped body. Hence it is obvious that its action at any point (x, y, z] will lie in the plane which passes through this point and the axis of z. It is assumed that while we keep to the same plane, this action will pass through a fixed point ; so that, denoting the co-ordinates of this point by f, 77, 0, we have !==??= x~ y~r^ where c is a constant quantity, and equal to VC? 2 -}- 77*), and r l stands for Put 5 for \/{(x f) 2 -f (y vY + z*}, and denote the action of the mass by //-s >n . Then, with the usual notation, and Y and /Tcari be similarly expressed. MAUPERTUIS. 65 Now *dtf~ 1 and ( x ) dx+( y -}dy + zdz = rdr cdr . \ rj \ rj Thus, finally, the equation to the surface of relative equi- librium is n+l = constant, n+l that is, ^- (x z -f 2/ 2 ) - = ^ = constant. 120. Maupertuis himself gives two investigations, one for the part of the mass which is between the axis of rotation and the point (f, 77, 0), and the other for the part which is beyond this point ; but this is unnecessary : a single investigation with proper generality in the symbols applies to the whole mass. The second problem includes the first as a particular case; we have only to suppose //, = 0. Maupertuis himself makes this remark : see his page 253. Maupertuis suggests, that the constants may happen to be so adjusted, that what we may call the generating curve of the ring will consist of two ovals ; so that, in fact, there will be two rings. This is conceivable, but he is wrong in implying that it is possible when m = 1, and n 1 ; for then the generating curve must consist of only a single ellipse. 121. The solutions here given by Maupertuis are reproduced by him in his Figure des Astres ; and also, though with less detail, in his memoir, which is published in the Paris Memoires for 1734. The problems, though rather theoretical than practical, were doubtless a valuable contribution to the science of Hydro- statics of the period. As to the popular part of the memoir, we shall say a word hereafter: see Art. 127. T. M. A. 5 66 MAUPERTUIS. 122. We have next to consider the work published by Maupertuis, under the title of Discours sur les difftrentes figures des Astres...Psiris, 1732. I have seen only the copy in the library of the Royal Society, which is marked Ex dono Auctoris. The volume is in octavo, and contains 83 pages, besides the Title and Table cf Contents, on four pages. The mathematical part of the volume consists of the same problems in French as were given in Latin in the Philosophical Transactions, and which we have already noticed. Besides this, we have Chapters of a popular character, which contain general reflexions on the figure of the Earth, a metaphysical discussion on attraction, and explanations of the motions of the planets on the system of vortices, and on the system of gravitation. 123. In his first Chapter, Maupertuis adverts to the re- searches of Huygens on the figure of the Earth, and afterwards to those of Newton. By taking this order, a reader might be led to suppose that Huygens preceded Newton in this subject ; but, as we have already pointed out, Newton was the first : see Art. 65. 124. There is a note on page 44 which presents a difficulty. Suppose a sphere, the radius of which is one foot, and its density the mean density of the Earth. The attraction which this sphere would exert on a particle at its surface, is a very small fraction of the attraction which the Earth would exert on a particle at the surface of the Earth ; the numerator of the fraction would be unity, and the denominator the number of feet in the Earth's radius. This substantially agrees with Maupertuis. Then he proceeds thus : " Deux Spheres semblables, placets a la distance d'un quart de pouce dans le vuide, employeroient un mois a se joindre." I suppose the spheres to be such as have been just mentioned, namely, each of a foot radius and of the mean density of the Earth ; and that they are to be placed so that their surfaces may be a quarter of an inch apart. But then instead of a month the spheres would require only a few minutes to arrive at contact. Thus I am quite at a loss as to his meaning. 125. A second edition of the Figure des Astres was published, which I have not seen. Clairaut refers to it on pages 19 and 59 MArPERTUIS. 67 of his Figure de la Terre ; see also D'Alembert's Opuscules Mathe*- jtHttiques, Vol. vi. page 358. The work seems to have been translated into English. 126. There is an account of Maupertuis's Figure des Astres on pages 85... 93 of the historical portion of the volume of the Paris Memoir es for 1732. Centrifugal Force has puzzled the writer of the account; he says on page 86, " ...les directions de la Force centrifuge sont a chaque instant les Tangentes de chaque point...." Of course instead of tangents we ought to read normals. 127. The popular part of the Figure des Astres is reproduced in the collected edition of the works of Maupertuis, published in four volumes at Lyons in 1756 ; it occupies pages 81... 170 of the first volume. The mathematical investigations are not reproduced. Maupertuis suggests that the variable brightness of certain stars may be explained by supposing that these stars are very much flattened, and that, owing to different positions assumed by their axes of rotation, we sometimes have a much larger disc turned towards us than at other times. He considers that the nebulae are really suns or planets, of figures more or less deviating from spheres. He suggests that the ring of Saturn may have been formed out of the tail of a comet which Saturn by the aid of his at- traction has appropriated. 128. A memoir by Maupertuis, entitled Sur les loix de I Attraction, is contained in the volume for 1732 of the Paris Memoires, published in 1735. The memoir occupies pages 343. . .362 of the volume. There is an account of the memoir on pages 112... 117 of the historical portion of the volume; this account, like many other attempts to give a translation of mathematical processes into ordinary language, is scarcely intelligible. The memoir, according to Bailly, is the first example of the adoption of the principle of attraction by French mathematicians: see Histoire de FAstronomie Moderne, Vol. ill. page 7. The memoir may be described as an analytical investigation of most of the results contained in Newton's two sections on Attraction ; adding, however, nothing of importance to them. The methods employed are simple and interesting. 52 68 MAUPERTUIS. 129. We will notice the method by which Maupertuis finds the attraction of a spherical shell. Suppose the law of attraction that of the inverse n ih power of the distance. Proceeding as in Art. 4 we obtain for the attraction of an element of the shell ^ cos 6. Now it will be found that - '- = -. ^ , and r n dr c sm ' y = r sin 6. Thus the expression becomes - -^ cos ; and a r* + c*-a* cos 6 = x so that finally we have 2cr which is immediately integrable. This is substantially the method of Maupertuis ; the chief part of it consists in making r the independent variable. The method is, in fact, that which Laplace adopted for finding the attraction of a spherical shell; and it has passed into the ele- mentary text-books on the subject : see Statics, Chapter xm. It will be noticed that in this process Maupertuis made the easy extension which arises from taking the inverse n th power of the distance ; while Newton, in the corresponding place, used only the inverse square of the distance. 130. Some incidental statements made in the memoir may be noticed. Maupertuis says on page 343, that a homogeneous fluid mass which has no motion of rotation, but is left to the influence of its own attraction, will necessarily assume a spherical form : " car il est facile de voir qu'il n'y a que cette figure dans laquelle toutes les parties puissent demeurer en equilibre." The belief here ex- pressed w.as doubtless held by many of the earlier writers on the subject ; but the belief was not founded on evidence. It is ob- served by Poisson that it has not been demonstrated that the sphere is the only figure which can be taken by a fluid at rest under the mutual attractions of its particles, however natural that may appear. Traite de Mecanique, Vol. n. page 543. See also Re*sal, Traite ttementaire de Mecanique Celeste, page 198. Maupertuis says on page 346, that if a homogeneous fluid rotates round an axis, and its particles are attracted towards a MAUPERTUIS. 69 centre by a force which varies as the distance, the form assumed is such that the meridians are ellipses : this we know to be true, with the condition, however, that the centre of force must be at some point of the axis of rotation. He adds with respect to the fluid mass: "Et si elle circule autour d'un axe pris au dehors d'elle, elle forme un anneau dont les sections sont encore des ellipses." This passage taken alone would not be intelligible, but from another memoir we know all that Maupertuis can have intended to say ; namely, that relative equilibrium will subsist under a certain peculiar assumption : see Art. 119. Maupertuis offers some remarks on his pages 347 and 348, commencing with the following sentence : " Suppose que Dieu eut voulu e'tablir dans la matie're quelque loi d' Attraction, toutes ces loix ne devoient pas lui paroitre egales." Maupertuis holds that the ordinary law has, as it were, a reason for preference, because it leads to the result that a sphere will attract as if it were a particle collected in its own centre. To this Stay alludes in his Philosophies Recentioris, Lib. IV. v. 1582... 1584: Scrutantes quidam ; quid Muncli illexerit ipsiim Artificem, legem ut voluisset material Ponere, quam doceo ; . . . Boscovich in his note dissents from Maupertuis. See also Bailly, Histoire de r Astronomic Moderne, Vol. ill. page 7. Maupertuis refers on page 361 to thirty -propositions relating to attractions, given at the end of Keill's works ; and on page 362 he says that Keill and many English philosophers believed preci- pitations, coagulations, crystallizations, and a multitude of other phenomena to arise from an attraction very powerful at contact, but insensible at great distances. He adds : " Enfin M. Friend a donne 7 une Chimie, toute de'duite de ce principe." 131. In the Paris Mtmoires for 1733, published in 1735, we have a memoir by Maupertuis, entitled Sur la Figure de la Terre, et sur les moyens que I' Astronomic et la Geographic fournissent pour la determiner. The memoir occupies pages 153. . .164 of the volume. Maupertuis gives analytical investigations of the length of a degree of longitude and of a degree of meridian on the Earth, supposed to be an ellipsoid of revolution ; and he shews how the 70 MAUPERTUIS. axes of the ellipsoid may be deduced from lengths of degrees determined by measurement. Maupertuis refers to Huygens, Newton, Cassini, Mairan, and M. des Aiguiliers ; the last is usually written Desaguliers. Maupertuis also quotes a passage from a letter written by Poleni : we shall notice the letter in Chapter VIII. 132. In the Paris Mtmoires for 1734, published in 1736, we have a memoir by Maupertuis, entitled Sur Us Figures des Corps Celestes. The memoir occupies pages 55... 109 of the volume; there is an account of it on pages 88... 94 of the historical portion of the volume. The memoir may be regarded as a development of the Figure des Astres ; for Maupertuis says on page 56 : Je reviens a examiner les figures que les loix de la Statique et de PHydrostatique doivent dormer aux Corps celestes, et j'entrerai sur cette matie"re dans un plus grand detail que je n'ai fait dans le Discours sur la figure des Astres. The memoir is divided into four parts. 133. The first part of the memoir treats on a subject which Bouguer discussed in the same volume ; and adds nothing fresh. Maupertuis shews, as Bouguer did, that if the force on a fluid is always directed to i fixed point, the principles of Newton and of Huygens lead to the same form for equilibrium, provided the force be a function of the distance from the fixed point ; but they do not lead to the same form if the expression for the force be the product of a function of the distance into a function of the angle which determines the position of the distance. 134. Maupertuis gives an extract of a letter sent to Fermat by Pascal and Roberval, in order to shew that the idea of attrac- tion had occurred to the writers before Newton proposed it. But we have here only a vague idea, not any suggestion of the law of the inverse square ; and of course no pretence at demonstration. 135. In the second part of the memoir we have the problems already given in the Philosophical Transactions ; though they are here treated with less detail : see Art. 121. .MAfl'KRTCJS. 71 For a particular case of the second problem, Maupertuis sup- poses that the force which is directed to a fixed point in the axis of rotation varies inversely as the square of the distance, and that the other force vanishes. His result then, expressed in modern notation, becomes - + ^ r* cos 2 = constant. This is in fact the equation which is now obtained in investi- gating the form of the atmosphere. Maupertuis does not discuss the equation ; but he implies that it would give him an oval curve about some point not coinciding with the pole from which r is measured. This, however, is not the case ; that is to say, the equation does not correspond to the diagram he supplies, and has no application to such an object as Saturn's ring, which he has in view. 136. In the third part of the memoir, Maupertuis refers to certain celestial phenomena which he considers support his theory; such as nebulas and variable stars. 137. The fourth part of the memoir relates to the figure of the Earth supposed fluid, and taking the ordinary law of attraction. This may be described as a commentary on Newton's theory of the Figure of the Earth. Newton's process is developed clearly and correctly ; with the exception of one slight mistake. In Art. 20, we have stated that the attraction of a Certain ob- latum is approximately a mean proportional between the attrac- tions of a certain sphere and a certain oblongum. Maupertuis incautiously says that the attractions of these bodies are as their masses, and therefore the result which Newton affirms is tru e. We have already drawn attention to this mistake : see Art. 22. 138. Maupertuis obtains, as Newton did, the value ^- for the ^j ratio of the difference of the axes to the minor axis in the case of Jupiter ; see Art. 29. Then Maupertuis says on his page 96 : Comme cette difference est beaucoup plus grande que celle qui resulte des observations de M. Cassini, et que celle qui resulte des observations (le M. Pound, M. Newton conjecture que Jupiter est plus dense vers le plan de son equateur que vers les poles. Get execs de 72 MAUPERTUIS. densit^ feroit qne la colorane qui est dans le plan de 1'equateur, pour 6tre en Squilibre avec celle qui repond au pole, doit etre plus courte que cette Theorie ne la determine, et par consequent le diametre de 1'equateur differeroit moins de 1'axe, et son rapport a 1'axe approcheroit plus du rapport observe*. This extract shews in what sense Maupertuis understood a rather obscure passage in Newton ; but of course the explanation is not very satisfactory. If the fluid is not homogeneous, the whole investigation must be revised; and it will not be sufficient to consider merely the equilibrium of the polar and the equatorial columns. This passage in Newton seems to have been considered rather important by Maupertuis, for he had previously noticed it, namely, on his page 73. But this reference was not very appropriate ; because Maupertuis is there using, not the law of attraction of nature, but the hypothesis of a force directed to a fixed point. 139. On the whole, it does not seem to me that this long memoir by Maupertuis added anything to the current knowledge of the subject ; the commentary on Newton was perhaps the most valuable part. 140. In the Paris Me'moires for 1735, published in 1738, we have a memoir by Maupertuis, entitled Sur la figure de la Terre. The memoir occupies pages 98... 105 of the volume. Maupertuis investigates the expression for the radius of cur- vature of an ellipse in terms of the inclination to the major axis ; namely, in modern notation, .. This furnishes a (1 - e* sin 2 \) f very approximate expression for the length of a degree of the meridian : see his page 99. Maupertuis also solves a problem which we may thus enun- ciate : find at what point the change in the length of a degree of the meridian is most rapid. Let a be the measured length of a degree in the latitude <, and let p be the radius of curvature; then we take -= - r 9 so p 180 MAUPERTUIS. 73 that T = jjjj p. Therefore g = ^ ^ . Hence J measures the rate of increase of the length of a degree ; and so we have to make --. a maximum. This is substantially the process of d(f) Maupertuis; see his page 105. The result is that <f> must be found from the equation 3e 2 sin 4 <j> - (4e 2 - 2) sin 2 </> 1 = 0. 77" If e is very small, we have approximately <f> = - . Maupertuis makes some simple remarks on the important subject of comparing the measured lengths of degrees of the meridian in the most advantageous manner, so as to render the gradual change in the length decidedly obvious in spite of the unavoidable errors of observations. See his pages 101... 104. 141. In the Paris Memoires for 1736, published in 1739, we have a memoir by Maupertuis, entitled Sur la Figure de la Terre ; the memoir occupies pages 302... 312 of the volume. Maupertuis suggests the following operation. Take two stars which have about the same right ascension and a difference of one degree in declination. Find two places A and B on the Earth's surface, such that one of these stars passes over the zenith at A, and the other over the zenith at B. Then determine by measure- ment the place G on the Earth's surface, which is on the arc AB, and equally distant from A and B\ and observe at C the zenith distances of the two stars. If the Earth is a sphere these zenith distances ought to be equal ; if the zenith distances are not found to be equal, we have evidence that the form is not spherical, and we have information as to whether it is oblate or oblong. Maupertuis also considers an important point in connexion with a trigonometrical survey; namely, the ultimate effect of a constant cause of error by which each side of the triangles em- ployed in succession to produce the required result is rendered greater than it should be. Then, combining this with the error which may be expected to arise from the astronomical observations for finding the amplitude of the arc, he determines what he considers to be the most advantageous number of triangles to be employed. <4 MAUPEKTU1S. Maupertuis refers to this memoir in his account of the opera- tions in Lapland ; for there the conditions which, according to the memoir, are most advantageous were reasonably satisfied. See his work La /'/'////</ elf la Terre detcniihu'e, page. '>">. 142. The Paris Mi'inoircs for 17'>7, published in 1740, contain on pages oN})...4(>(> a memoir by Maupertuis, entitled La Fiyure de la Terre deter mine'e...; the memoir describes the operations in Lapland which established the oblate form. There is an account of the memoir on pages <)()...DG of the historical portion of the volume. The memoir isembodied in the book which Maupertuis published in 17o3 under the same title: we shall notice this hereafter. 143. A book was published in 17o<S, entitled h\c(it/ien (h's- interesse des dijferens oiivratjes <^ui out ete f<nt* pour determiner la lifjure de la terre. See La Lande's Btbhuyrapltie Astrononin^u& ) page 40(5. La Lande says tliat this book is marked Olt/e/t!>o/<ry, but was ]rinted at Parix: he adds, that owing to the censorship of the press a book was often marked with the name ot some supposed place where the press was free, as London or Amsterdam. I have not seen this edition. La Lande on his next page gives the title of another work pub- lished in 17 "to and also marked Oldenbotirg, namely, E.cnincn des troit dissertations <j_ne M. Desagidiers a publu'es sur l<t fit/lire dc la t'jrre, dcins Ics Transactions PltilosopJiif^ues, J\o,y. *>NG, *>S7 ct oSS. I have not seen this edition. 144. r rhc' two works appear together in one volume which is dated 1741, and marked Amsterdam; this volume 1 will now describe. The volume is in octavo ; there are forty-six unnumbered pages, followed bv KiO which are numbered. The Kjcameti dr'&intet'essd extends to page 1 1 ; mid the rest of the volume is devoted to the I'jjcdinan tic.') ti'ois disxertdttijHS. !!-.">. I begin with the I'^rann'ii df'siittf'/'C'twt'. The title-page says that it is the second edition, augmented ly the history of the book. The title-page has the motto, " Kt munduin tradidit lisjtutationi eorum. Kccles. cap. III. v. 11." MAUPERTUIS. 75 140. The work is anonymous ; but La Lande says that it was written by Maupertuis. This is also clear from other sources. See Bouguer's Figure de la Terre, pages 174 and 175, and his Lettre...Astronomique Pratique, pages 6, 7, 9, and 10; also La Condamine's Reponse...p&ge 5. It affects to be very impartial, and is certainly very clever and amusing; but it contributes nothing new to the knowledge of the subject. The work seems to have attracted great attention at the time ; and, as we learn from the Introduction, it was attributed to Mairan and to Fontenelle, although they were opposed to the opinion of Maupertuis. In fact, as La Lande remarks, the smart bantering tone of the work might easily deceive a reader and leave him doubtful whether the author was in favour of the oblate or oblong form. 147. Thirty-six of the unnumbered pages are devoted to the Histoire du Livre ; these pages constitute an outline of the con- tents of the work. But one matter here considered is not in- cluded in the work ; it had, I presume, happened since the publi- cation of the first edition. A distinguished Danish astronomer, named P. Horrebow, had written a work on the Theory of the Earth, and well-feigned surprise is expressed at his rashness in declaring for the oblate form. See Petri Horrebowii Opera, 1740, Vol. I. page 381. 148. In the first part of the work the writer speaks of the important measurements which had been made, namely, that at the polar circle which favoured the oblate form, and five opera- tions by Cassini which favoured the oblong form. The measure- ment at the polar circle will be discussed in Chapter VII. In noticing the operations at the polar circle the writer puts the amplitude of the arc at 57' 25", omitting the correction for Aberration which he says is not yet allowed by all the world. By omitting the Aberration the two determinations of the amplitude, by two different stars, agree to a second. The length of the degree is first stated as 57437 toises ; but this is the length which Maupertuis really obtained by allowing for Aberration, and is, I presume, a misprint. Afterwards, the number is given as 57497 ; and this is what it should be if we neglect Aberration. 76 MAUPERTUIS. The five operations by Cassini are those which are described in the Paris Mtmoires for 1701, 1713, 1718, 1733 and 1734 The writer says in his usual jesting manner that since all these opera- tions were in favour of the oblong form he is astonished that any more should be sought ; and he often recalled the saying of an ancient, that if ignorance is the punishment of too little study, uncertainty is often the reward of too much. 149. In the second part of the work, we have notices of the authors who had discussed the theory of the Figure of the Earth. For the oblate form Huygens, Newton, David Gregory, and Hermann are brought forward. For the oblong form the far less eminent names of Childrey, Burnet, Eisenschmidt and Mairan are brought forward. Childrey seems to have been the author of a description of England ; the others we have already mentioned. 150. Let us now turn to that part of the volume which is devoted to the consideration of the dissertations published by Desaguliers. The title-page has the motto : Magnus sine viribus ignij Incassum furit. VIRG. Georg. Lib. III. v. 99, 100. There is no statement that this is a second edition ; it is dated 1741. After the title, we have a notice by the bookseller ; he ascribes the work to a learned friend to whom he had shewn Ihe former work. La Lande does not say by whom it was written, but I presume that the whole volume is really by the same author, that is, by Maupertuis. The work shews that some of the objections which Desaguliers had brought against Cassini were really unfounded ; especially those in the first of the three dissertations. It will be made clear , hereafter, that Desaguliers was not judicious in his criticisms : see Chapter VIII. CHAPTER V. STIRLING. 151. STIRLING was the first person who turned his attention to the important point which had been assumed by Newton in his theory of the Figure of the Earth ; see Art. 44. The memoir which we shall now notice is entitled, Of the Figure of the Earth, and the Variation of Gravity on the Surface. By Mr. James Stirling, F.E.S. The memoir occupies pages 98... 105 of Number 438 of the Philosophical Transactions, which is for the months July, August and September, 1735. The Number forms part of Vol. XXXIX. which is for the years 1735, 1736, and is dated 1738. 152. Stirling begins thus : The Centrifugal Force, arising from the Diurnal Rotation of the Earth, depresseth it at the Poles, and renders it protuberant at the Equator ; as has been lately advanced by Sir Isaac Newton, and long a & hy Polybius, according to Strabo in the Second Book of his Geo- graphy. But although it be of an oblate spheriodical Shape, yet the kind of that Spheroid is not yet discovered ; and therefore I shall suppose it to be the common Spheroid generated by the Rotation of an Ellipsis about its lesser Axis ; although I find by Computation, that it is only nearly, and not accurately such. I shall also suppose the Density to be every where the same, from the Center to the Surface, and the mutual Gravitation of the Particles towards one another to decrease in the duplicate Ratio of their Distances. The late Sir J. W. Lubbock says in the Preface to his Account of the " Traite* sur le Flux et Efflux de la Mer " of Daniel Bernoulli : I have searched in Strabo in vain for the remarkable passage alluded to by Stirling ; but at all events the glory of the discovery of the true figure of the Earth belongs to Newton. 78 STIRLING. Perhaps Sir J. W. Lubbock expected too much. Strabo certainly says that Polybius supposed the equatorial regions to be elevated. See page 97, near the bottom, of Casaubon's edition of Strabo, the paging of which is given in the margins of other editions. See also a note on page 254 of Vol. I. of the French translation of Strabo by De la Porte du Theil and Coray. 153. Stirling states without demonstration approximate re- sults respecting a homogeneous oblatum. He gives the direction and the magnitude of the action which the oblatum exerts on a particle at its surface, both when the oblatum does not revolve, and when it does. The approximations are true to the order of the square of the excentricity of the generating ellipse. Let P.- denote any point on the generating ellipse ; let CA and CB be the semi-axes. Let PG be the normal at P, meeting the greater axis at G. Take -?Cft Then Stirling says, when there is no rotation PH is the direction of gravity and proportional to the value of it. Draw PM perpendicular to CA\ let CM=x, and PM=y; let X and Y denote the attractions at P parallel to CA and CB respectively. Then if p denote the density, and e the excentricity, we have by the modern theory ZTrp(l-e*}x 1 2 sin 8 6 (1 - e 9 sin 2 J o d6, STIRLING. F = larpy \~ cos 2 sin 6 (1 - e* sin 2 0)- 1 dB ; * see Statics, Chapter xin. If we neglect e* and higher powers of e we shall obtain approximately. But by the nature of the ellipse (76r = e z x, so that Thus the component attractions may be represented by PM and MH in magnitude and direction ; and therefore the resultant may be represented by PH in magnitude and direction. When there is rotation and relative equilibrium PG represents the resultant action in magnitude and direction. Stirling does not make any distinction in language corresponding to the fact that this statement is exact while the former is approximate. We know that for the equilibrium of a fluid, the resultant action must be normal to the surface, so that PG is exactly the direction of this action. Now take the expressions given for X and F, and introduce the centrifugal force ; then the actions at P parallel to the axis of y and x respectively will be ny and \x, where p and X are constants : so that these actions are proportional to y and - x respectively. But as we know that PG is the direction of the resultant, the components must be proportional to PM and M G, respectively ; hence MG must be equal to - x, and PG will represent the resultant in magnitude and direction. This simple process does not occur very often in works on the subject: it is given on page 113 of Laplace's Theorie...de la Figure elliptique des Planetes, at least substantially. 80 STIRLING. 154. Let X denote the latitude of P, that is, the angle PGM: this will of course be very nearly equal to the angle PCM. We can express PH and PG in terms of A, and the elements of the ellipse ; thus we obtain the following approximate results which in effect Stirling gives : first suppose no rotation, then if F denote / e * \ the attraction at the pole, the attraction at P is Fl 1 cos 2 XJ ; next suppose rotation, then if G denote the gravity at the pole, / e 2 \ the gravity at P is G ( 1 x- cos 2 X j . In the diagram of Art. 153, the attraction at P is denoted by PH, and the gravity at P by PG : thus, as Stirling remarks, HG represents the centrifugal force at P. It is easy to give exact statements of the nature of Stirling's approximations; this, as we shall see hereafter, was done by Thomas Simpson. 155. Stirling applies the expression for the value of gravity at any point of the surface to some observations respecting the relative number of vibrations of the seconds pendulum at London and at Jamaica ; he deduces from these observations as the _L 7 J_ ellipticity : but he goes on to shew that this value is inadmissible. Stirling makes the following remark respecting pendulum observations : From all the Experiments made with Pendulums, it appears that the Theory makes them longer in Islands, than they are found in fact... This Defect of Gravity in Islands is very probably occasioned by the Vicinity of a great Quantity of Water, which being specifically lighter than Land, attracts less in Proportion to its Bulk. Modern writers however appear to suggest that gravity may be greater on islands than on continents : see Airy's Figure of the Earth in the Encyclopaedia Metropolitan^ page 230, and Stokes's Variation of Gravity at the Surface of the Earth in the Cambridge Philosophical Transactions, Vol. vin. 156. We have seen in Art. 44, that Newton assumed without demonstration an oblatum as a possible form of relative equili- brium for a mass of revolving fluid. Laplace asserts that the STIR LI XU. SI defect was first supplied by Clairaut in the Philosophical Trans- actions for 1737 ; see the Mecanique Celeste, Vol. v. page 6. But perhaps we may consider that Stirling had already obtained this result. The main thing to be proved was that the resultant action at any point of the surface would be normal to the surface, when a proper relation was established between the ellipticity and the ratio of the centrifugal force to the attraction. The re- \ GM lation, in the notation we have used, is that - = -^i> : that is, //. L>M . - = 1 e*. I do not say that Stirling gives this relation explicitly; but it seems to me implied in his remarks. Such too appears to have been the opinion formed at the time ; as we may infer from a passage in the Philosophical Transactions, Vol. XL. page 278, which will be quoted in Art. 168. See also Lubbock, Account of the Traite. . ., page vi. However, Stirling's results were given with- out demonstration ; moreover, we find from the passage in the Philosophical Transactions, to which reference has just been made, that they could not have been known to Clairaut when he wrote his first paper on the subject; so that Clairaut's merits remain undiminished. 157. I find it difficult to ascertain what opinion Stirling held as to the agreement of the theory with facts. He says, as we have seen, in his commencement referring to the Earth's elliptic figure, "that it is only nearly, and not accurately such." But further on he says very positively : And whereas the Earth could not be of an oblate spheroidical Figure, unless it turued round its Axis; nor could it turn round its Axis, without putting on that Figure... Moreover he compares his theory with observation in the case of Jupiter, and finds them to agree nicely ; then he says : And if this Theory agrees so well with Observations in Jupiter, there is no doubt but it will be more exact in the Earth, whose Dia- meters are much nearer to Equality. After he has made the suggestions respecting pendulum ob- servations on islands, which we have quoted, he gives the following statements : And I find by Computation, that the Odds in the Pendulums be- T. M. A. 6 82 STIRLING. twixt Theory and Practice is not greater than what may be accounted for on that Supposition. I shall also observe, that although the Matter of the Earth were entirely uniform, yet the Hypothesis of its being a true Spheroid is not near enough the Truth to give the Number of Vi- brations which a Pendulum makes in twenty-four Hours. He concludes thus : But after the French Gentlemen who are now about measuring a Degree, and making Experiments with Pendulums in the North and South, shall have finished their Design, we may expect new Light id this Matter. 158. Stirling's mathematical powers were highly esteemed by his contemporaries. Clairaut calls him " one of the greatest Geometricians I know in Europe." Philosophical Transactions, Vol. XL. page 278. See also Maclaurin's Fluxions, page 691 ; Todhunter's History of the Theory of Probability, pages 188, 190. Stirling's name seems to be omitted in the ordinary biogra- phical dictionaries. The Abridgement of the Philosophical Trans- actions by Hutton, Shaw, and Pearson, contains some notices entitled Biography ; or, Account of Authors. All that is there recorded of Stirling is in Vol. VI. page 428, where we read : " This very respectable mathematician was agent for the Scotch Mine Company, Leadhills. He died the 5th of December, 1770." Sir John Leslie gives an interesting notice of Stirling in the Disser- tation on the Progress of Mathematical and Physical Science, which forms part of the Encyclopaedia Britannica : see page 711 in the eighth edition of the Encyclopaedia. CHAPTER VI. V CLAIRAUT. 159. IN this Chapter we shall give an account of certain memoirs by Clairaut; these exhibit the high mathematical power of their author, and form the origin of the researches afterwards embodied by him in his great work entitled Thforie de la Figure de la Terre. ICO. In the Paris Memoires for 1733, published in 1735, we have a memoir by Clairaut, entitled Determination geome'trique de la Perpendiculaire a la M&ridienne tracee par M. Cassini ; avec plusieurs Methodes den tirer la grandeur et la figure de la Terre. The memoir occupies pages 406... 416 of the volume. Clairaut shews that by such a process as Cassini adopted, the curve of minimum length between its extreme points on the surface of the Earth is obtained ; and this curve is not in general a plane curve, unless the Earth is a sphere. Clairaut then proceeds to investigations respecting curves of minimum length. For a surface of revolution he obtains the property, now well known, that the sine of the angle made by the curve at any point with the meridian varies inversely as the length of the perpendicular from the point on the axis of revolu- tion. He gives special attention to the case in which the surface is an ellipsoid of revolution. A mistake occurs on page 414, which also influences page 416. Clairaut says that if m is greater than unity u* 1 -I- - 5 is obviously 62 84 u~ 1 tc . greater than -.,; hut ?r 1 is a negative nuantitv, ami so w yr his statement is wrong. 1(1. Jn tin- Paris Mnmtli'L'fi for 17o-">, published in 17oS, we have a memoir by Clairaut entitled Sur Id iminrlle Nf'thude de JA. Ca*stni, jnni.r ca/i/tn/t/'e la Pujui'e de l<! Tcri'c. The memoir occupies pages 117.. 1-- of the volume. This memoir consists of simple and interesting investigations of tin- geometrical theorems involved in the application of Cassini's method. An important proposition in solid geometrv occurs here, per- haps for the first time. At any point, J/, of a surface of revo- lution, let a normal section be made at right angle> to the plane of the meridian ; then the radius of curvature of this section at J/ is the length of the normal between J/ and the axis of revolution. Clairaut's demonstration is sound; but he leaves to his readers the trouble of constructing a diagram \vithout any dhvrtimis. ~H>'2. In the Paris Mrmiiirrx for 17-><), published in 17oJ>, we have a memoir by Clairaut, entitled /v//- In, Mature dc la Terrc par pltisieurs Arrx de M<'ridien i>ri$ a difif'reutc.x T^itiimlex. Tlie memoir <jccu])ies ])ages 111... 1^0 of the volume. Let x be the abscissa and y the ordmate of anv ]>o)nt on a curve ; and suppose that the radius of curvature is equal to /-/ _i_ l^ | -j- <%!'"' -I- . . , whei'e A is the ani/le \\hose tangent is , . and t/ij a, 1>. c are constants. Then ('lairaut shews how wo may express ./ and i/ in terms of ,r, which denotes (lie sine of .1. He practical I v cotiJine> hnnx-ll 1<> the case in which the above series, contains onlv the throe term- explicitlv given ; and for this case he calculates soine numerical results which niiglil be useful for application to the arc- ahoiit to be measured in Lapland and Peru, compared with that measured in France.. Let /// denote the excess of the radius of curvature at the equator above that at latitude LY', and let p denote the excess of CLAIRAUT. 85 the radius of curvature at latitude 45 above that at 67 ; then 2551w +2904 Clairaut finds a --- *- for the equatorial semi-diameter, O-OD , and a + - for the polar semi-diameter. I have cor- oZoo rected a sign in the former value. On the Cassinian hypothesis m and p will both be positive, on the Newtonian hypothesis they will both be negative. 1C3. We have next to consider a memoir by Clairaut entitled Investigations* aliquot, ex quibus probetur Terrce figurant secundum Leges attractionis in ratione inversd quadrat* distantiarum maxime ad Ellipsin accedere debere, per Dn. Alexin Clairaut, Eeg. Societ. Lond. et Eeg. Scient. Acad. Paris. Soc. This memoir occupies pages 19... 25 of Number 445 of the Philosophical Transactions; which is for the months January... June, 1737. The Number forms part of Vol. XL. which is for the years 1737, 1738, and is dated 1741. The object of the memoir is to demonstrate Newton's postulate ; see Art. 44. Clairaut obtains an approximate expression for the attraction of an oblatuin at any point of its surface; and thus shews, that with a suitable value of the ellipticity the resultant of the attraction and centrifugal force at any point of the surface will be normal to the surface at that point. 164. In Clairaut's work on the Figure of the Earth he did not reproduce this approximate solution of. the problem of the homo- geneous oblatum; for Maclaurin had in the meantime given an exact determination, of the attraction of such a body, and so Clairaut followed him and exhibited an exact solution : see Clairaut's Figure de la Terre, page 157. But the method used in this memoir for trie homogeneous oblatum is used in the work for the heterogeneous oblatum; pages 233... 243 of the work reproduce the essence of this very ingenious method. 165. In this memoir we have for the first time the approxi- mate method of determining the attraction of an oblatum on a 86 CLAIRAUT. particle at its pole, which still retains a place in elementary works : see Statics, Art. 217. The method occurs in the Figure de la Terre, pages 239... 2 43, where it is used for a particle situated at any point of the polar axis produced. 166. We may observe that Clairaut's memoir begins rather inauspiciously by apparently adopting the error we have noticed in Newton and David Gregory : see Arts. 33 and 84. However, as we proceed we find that Clairaut really understood the theorem correctly : see especially page 24 of the memoir, and also pages 188... 190 of the Figure de la Terre. 167. The next memoir is entitled, An Inquiry concerning the Figure of such Planets as revolve about an Axis, supposing the Density continually to vary, from the Centre towards ike Surface ; by Mr. Alexis Clairaut, F.R.S. and Member of the Royal Academy of Sciences at Paris. Translated from the French by the Eev. John Colson, Lucas. Prof. Math. Cantab, and F.R.S. This memoir occupies pages 277... 306 of Number 449 of the Philosophical Transactions, which is for the months August and September, 1738. The Number forms part of Vol. XL. 168. Clairaut begins by adverting to Newton's researches on the Figure of the Earth, and especially to his important postulate; see Art. 44. Clairaut says : What at first seem'd to me worth examining, when I apply'd my- self to this Subject, was to know why Sir Isaac assumed the Conical Ellipsis for the Figure of the Earth, when he was to determine its Axis I began then with convincing myself by Calculation, that the Meri- dian of the Earth, and of the other Planets, is a Curve very nearly approaching to an Ellipsis ; so that no sensible Error could ensue by supposing it really such. I had the Honour of communicating my Demonstration of this to the ROYAL SOCIETY, at the Beginning of the last Year ; and I have since been inforin'd, that Mr Stirling, one of the greatest Geometricians I know in Europe, had inserted a Discourse in the Philosophical Transactions, No. 438. wherein he had found the same thing before me, but without giving his Demonstration. When I sent < LAIHAUT. 87 that Paper to London, I was in Lapland, within the frigid Zone, where I could have 110 Recourse to Mr Stirling's Discourse, so that I could not take any Notice of it. Of course Clairaut did not demonstrate, as he says, that the meridian is nearly an ellipse, but only that an ellipse is an ap- proximate solution. As we have stated in Art. 180, the earlier writers often assumed that a fluid mass, if acted on by no external force, would necessarily assume a spherical form. In like manner, when Newton's postulate had been established, it was often as- sumed, as here implicitly by Clairaut, that a fluid mass rotating with uniform angular velocity, and in relative equilibrium, would necessarily assume the form of an oblatum. 169. The first part of the present memoir determines the attraction at any point of an ellipsoid of revolution, supposing it to be composed of similar strata varying in density. The inves- tigations are only approximate, extending to the first power of the ellipticity. All that this part of the memoir contains is included in Clairaut's Figure de la Terre; but in the work there is a gain both as to simplicity and to generality. Problem I. of the memoir cor- responds to Section 45 on pages 239... 243 of the work. Pro- blem II. and Problem III. of the memoir are included in Section 46 on pages 243... 247 of the work. The Theorem on page 282 of the memoir corresponds to Section 44 on pages 236... 239 of the work. Problem IV. of the memoir corresponds to Sections 24 and 25 on pages 200... 202 of the work. Problem V. of the memoir corresponds to Section 26 on pages 203... 208 of the work ; the investigation is given at full in the work, but only the ilt in the memoir. Problem VI. and Problem VII. of the me- moir are included in Section 29 on pages 209... 218 of the work. The work is more general than the memoir. In the memoir it is assumed that the strata are similar, so that the ellipticity is the same for all the strata ; in the work this is not assumed. In the work the formulae contain a general symbol to represent the density; in the memoir a law of density is assumed, the density being denoted by /}'' + gr q , where/ g, p, q are constants, and r is 88 CLAIKAUT. the variable polar semi-axis of the strata : the integrations are effected in the memoir, but the formulae are thus rendered less simple in appearance than they are in the work. 170. The second part of the memoir contains the application of the first part, to find the figure of a nearly spherical fluid mass which rotates about an axis. This part is unsatisfactory, because the only condition of equi- librium which Clairaut regards is, that the resultant action at every point of the free surface shall be normal to the surface at the point. This is not sufficient for the equilibrium of a hetero- geneous fluid mass. Clairaut discovered* his error, and acknow- ledged it; see page 155 of his Figure de la Terre: here he allows that his investigations in the memoir are untenable, except on the supposition that the interior parts of the Earth had been originally solid. In the Sections 37 and 39, on pages 225, 226, 228, and 229 of the work, we have an equivalent for pages 288.. .291 of the memoir, but expressed more accurately. 171. On page 294 of the memoir, we have the first appear- ance of the theorem which is now known as Clairaut's Theorem : see Section 49, on pages 249, 250 of the Figure de la Terre. We will state the theorem. From the value of gravity at the pole subtract the value of gravity at the equator, and divide the re- mainder by the value of gravity at the equator ; this fraction we shall call Clairaut's fraction. Then Clairaut's Theorem asserts that the sum of the ellipticity of the surface and Clairaut 's fraction is equal to twice the ellipticity of the Earth considered as a homo- geneous fluid. We shall defer the demonstration of the theorem until we give an account of Clairaut's Figure de la Terre. 172. Clairaut deduces from his theorem a result contrary to a statement made t>y Newton ; see Art. 30. Clairaut, speaking of Newton, says : He affirms, that the Earth is denser towards the Centre than at the Superficies, and more depress'd than his Spheroid requires. But by the foregoing Theory we may easily perceive, that if the Density of the Earth diminishes from the Centre towards the Superficies, the Dimiuu- CLAIRAUT. 89 tion of Gravity from the Pule towards the Equator will be greater than according to Sir Isaac's Table ; but at the same time the Earth will not be so much depress'd as his Spheroid requires, instead of being more so, as he affirms. The two statements made by Clairaut are connected by his Theorem, so that one will follow from the other. In the Section 38, on pages 226, 227 of his work, he shews that if the density di- minishes from the centre to the surface, the ellipticity is in general less than for the homogeneous body: the condition which prevents the statement from being universally true is there given. Clairaut proceeds to say : Yet I would not by any means be understood to decide against Sir Isaacs Determination, because I cannot be assured of his Meaning, when he tells us, that the Density of the Earth diminishes from the Centre towards the Circumference. He does not explain this, and perhaps instead of the Earth's being compos'd of parallel Beds or Strata, its Parts may be conceived to be otherwise arranged and disposed, so as that the Proposition of Sir Isaac shall be agreeable to the Truth. In his Figure de la Terre, however, Clairaut does not hesitate to decide against Newton : see Art. 30. 173. As an example, Clairaut takes the following case : Setting aside all Attraction of the Parts of Matter, if the Action of Gravity is directed towards a Centre, and is in the reciprocal Ratio of the Squares of the Distances, the Ratio of the Axes of the Spheroid will then be that of 576 to 577 : And the Gravity at the Pole is greater than at the Equator by y-j-j th Part, or thereabouts. Which may be a Confirmation of what is here advanced, especially to such as will not be at the Pains of going through the foregoing Calculations. For we may consider the Spheroid now mention'd, in which Gravity acts in a reciprocal Ratio of the Squares of the Distances, as composed of Matter of such Rarity, in respect of that at the Centre, that the Gravity is produced only by the Attraction of the Centre or Nucleus. This is the first appearance of a problem which may be de- scribed as a companion to that discussed by Huygens ; and which has sometimes been erroneously ascribed to Huygens: see Art. 64. y() CLAIKAUT. 174. Clairaut makes some remarks on the two principles which were then in use for determining the form of a fluid in equili- brium, namely, Newton's principle of balancing columns and Huygens's principle of the plumb-line: he states the reasons which induced him to adopt the latter principle. He proceeds to exa- mine whether the solution which he has obtained does make the polar and equatorial columns balance ; he finds that, in order to secure this, a certain relation must hold among the constants which enter into the expression for the density. In fact, as we have already stated, Clairaut' s solution in the memoir did not satisfy all the necessary conditions: see Art. 170. 175. Clairaut demonstrates a result on pages 302... 304 of the memoir, which though quite obvious on the modern theory of fluid equilibrium must have appeared remarkable at the time. We will state the general proposition of which his result is a particular case. Suppose a solid, not necessarily homogeneous, covered with a stratum of homogeneous fluid which is in equili- brium ; then if a fine channel be made in the body from one point of the fluid to another, and be filled with the fluid, the fluid in the channel will remain in equilibrium. In fact, the pressure p at any point of the channel of fluid can theoretically be found so as to satisfy the necessary conditions. 170. The memoir closes with some reference to the results obtained by observations. Clairaut admits that those furnished bv the expedition to Lapland do not agree well with the theory; for, according to these, each of the two fractions which occurs in Cluiraut's Theorem .is greater than _ However he will wait for the observations made in Peru. 177. In the; Paris Ib'mmrex for 173!), published in 1741, there is a memoir by ( lairaut, entitled Suite (Tun ^[e.moire donne" >// 17-)->; (jiii a i>nr tit re: Determination Geomtftrique de la Pcr- jH'udi'-iiftiire r) Id M< f /-i<!teiine, <(<:. The memoir occupies pages S3... DO of the volume. In modern langungo we should say that this memoir relates 1> c-eode^ic curves on the. surface of an ellipsoid of revolution. CLAIKAUT. 91 The investigations are approximate, extending to the first power of the ellipticity. It may be interesting to give a specimen of Clairaut's inves- tigations. Let the polar semi-diameter be taken for unity, and let - iii> denote the equatorial semi-diameter. Let x denote the longitude of any point in a geodesic curve, measured from the meridian which the geodesic curve crosses at right angles ; let t denote the cotangent of the latitude of this point; let p denote the value of t when x = ; then dx _ pm V(l + f) dt~ 't^+~m Olairaut established this formula in his memoir of 1733 ; and it may be easily obtained from well-known works on solid geometry. Now put m = 1 a, and suppose a so small that its square may be neglected ; thus we get dx _ p apt dt ~ t*j(f-p*) " (T+7 2 ) -i s Clairaut does not use the symbol sin" 1 ; but he proposes the symbol As to denote what we denote by sin~ l s. The equation (1) determines x when t is known. Now Clairaut proceeds to determine t from it when x is known ; and for this he employs a special process, which we will now explain. Suppose that t r + AT, where T is the value of t which would correspond to the known value of x when a is zero, and so AT is very small. Hence from (1) we get . siu T^p _P + ' (2). But by supposition x = sin" 1 Hence, neglecting tlie term which involves the product of a and AT, we have lYoni ''!) P A */> , -i \ 7 ^ 2 -//'I TN/(r 2 -/) V(1+/>*) S1 VU+TV r l lii> furnishes the correction AT, which will he required in the cotangent of the latitude when calculated for a sphere, in order to obtain the value for the ellipsoid of revolution. Clairaut himself uses t for our T, and dt tor our AT. Clairaut's memoir consists of the solution of four problems; the other three re.^emblc that which we have taken as a specimen. They are illustrated by numerical application to an oblatum in which 1= - this value Clairaut says docs not differ much from that obtained by means of the degree of the meridian mea- sured at the p>lar circle. This memoir is the bust of the series of Clairaut's Contributions t" our subject before the publication of his work entitled Tln-orie ilc /'/ /'///"/' tlv l<i Tcrre, which we shall examine in Chapter XL: we now proceed to give an account of the measurement in Lap- land, to which allusion has jus-t been made. CHAPTER VII. ARC OF THE MERIDIAN MEASURED IN LAPLAND. 178. THE Academy of Sciences at Paris seems to have selected the problem of the Figure of the Earth as peculiarly its own. But the success hitherto attained scarcely corresponded to the labour which had been expended; partly perhaps owing to the fact that the able observers, trained by the astronomers who bore the justly celebrated name of Cassini, had adopted the oblong form and maintained it firmly. In order to settle the question in dispute between the Cassinians and the Newtonians, the scheme was seriously proposed in 1733 of measuring an arc of the meridian near the equator, in order to compare the corresponding length of a degree with that which had been obtained from the French arc by Pfcard and by J. Cassini. The task was entrusted to three members of the Academy, Bouguer, La Condamine, and Godin, who started in May, 1735. Two Spanish naval officers, Juan and Ulloa, assisted in the work. 179. After this expedition had started for Peru it was re- solved to measure also an arc as near as possible to the pole: see La Condamine, Journal du Voyage... page 1. This task was entrusted to four members of the Academy, Maupertuis, Clairaut, Camus, and Le Monnier; moreover I'Abbe' Outhier, who was a correspondent of the Academy, and Celsius, who was professor of Astronomy at Upsal, were associated with the Academicians. 180. The Arctic expedition seems to me to have been stronger than the Equatorial. The genius of Clairaut outshone that of the 94 ARC OF THE MERIDIAN MEASURED IN LAPLAND. whole Academy, which was not yet adorned by the rising splen- dour of D'Alembert. But even if we leave out of consideration this transcendant name the superiority remains, I think, still with the Arctic party. I should place Maupertuis, Camus, and Le Monnier, above Bouguer, La Condamine, and Godin; while the priest and the professor who accompanied the former are at least equal to the two sailors who assisted the latter. The two operations were conducted on different principles. The members of the Arctic expedition worked in harmony under the general direction of Maupertuis. La Condamine calls Maupertuis, the senior (Canoien) of the party, Journal du Voyage... page iii. ; and Maupertuis is called Chef de I'mtreprise du Nord in the Histoire de FAcade'mie...foT 1737, page 96. There was but little cordiality in the Equatorial party; and the three Acade- micians performed much of their work separately. Thus in the former case we find friendship and subordination; and in the latter case isolation and independence. On a purely scientific estimate it may be maintained that there are advantages in each course which the other does not secure. We are here concerned only with the Arctic party which left Paris on the 20th of April, 1736. Two narratives of the proceed- ings were printed ; we will now describe these works. i 181. Maupertuis published La Figure de la Terre d&erminte par les observations .. .au cercle polaire. Paris, 1738. This is an octavo volume ; the Title, Preface, and Table of Contents, occupy xxviii. pages; the text occupies 184 pages; there are 9 plates besides a map. In the historical portion of the Paris Memoir -es for 1737, pages 90... 96 relate to the Arctic expedition: the date of publication is 1740. Moreover, in this volume, pages 1. .130 of Maupertuis's work are reprinted; they occupy pages 389... 465 of the volume. Maupertuis here says there have been too many editions of his book in various languages to render it necessary to repeat the other observations made in the North : he contents himself with referring to the observations on the force of gravity, and repro- duces the Table which occufs on page 181 of his book. ARC OF THi: MERIDIAN MEASURED IN LAPLAND. J),") It is stated by La Condamine that Maupertuis's work was translated into all the languages of Europe : Journal du Voyage. . . page iii. I Lave seen a German translation and a Latin transla- tion. The German translation was published at Zurich in 1741; it contains also a dedication to Frederic III. of Prussia, by Samuel Konig, an introduction by the translator, and a memoir by Celsius on Cassini's work De la Grandeur et de la Figure de la Terre. The Latin translation was published at Leipsic in 1742; it contains also an introduction by the translator, Alaricus Zeller : he says on the third page of his introduction that he has preserved the paging of the Amsterdam edition in his margin. This translator's introduction contains some criticisms which are not devoid of interest; they do not however practically affect the determination of the length of the degree of the meridian, but relate to inci- dental matters, such as refraction. There are also a few notes to the translation, which supply corrections of slight misprints or mistakes. There is an English translation which I have not seen. 182. Outhier published Journal d'un Voyage au Nord. . ., Paris, 1744. This is a quarto volume; the Half-title, Title, Dedication, and Preface, are on eight pages; the text occupies 238 pages, followed by two pages which contain an Extrait des Registres de I' Academic..., and the Privilege du Roi. According to the Table des Figures on page 238, there ought to be 18 plates. But in the single copy which I have seen there are only 16 plates. The plate which is marked 15 in the list does not occur; there is only one plate corresponding to the two which are marked 9, 10 on the list; and there are only two plates corresponding to the three which are marked 6, 7, 8 on the list. On the other hand, there is a Veue de la Montague de Niemi, du coti du Midy, which is not named in the list. Outhier's work seems never to have attracted much attention and to be now scarce. 183. The calculations and the theoretical deductions are given most fully by Maupertuis ; the details of the daily occupations of the party, and the peculiarities of the country and of the inhabit- :'<> A nc OF Tin: MERIDIAN MKAsrnr.n IN LAPLAND. ants, an.- given most fullv bv Outhier. I shall refer to the pages of Maupertuis in the original French edition, and distinguish them by the letter M. 1 shall refer to Outhier's work by the letter O. l>vl. Maupertuis was for a long time in doubt whether he should go to Iceland, to Norway, or to the Gulf of Bothnia; he decided for the last, intending to earrv on his operations among the islands along the shores of the Gulf. O. o. But on examina- tion these islands were found to be too low, and too near the shore, to form advantageous stations; and after some consideration Maupertuis resolved to proceed to the mountains north of Tornea, which is at the head of the Gulf. M. 1 1 ; O. :>:>. Finally Tornea was taken as the most Southern station, and Kittis as the most Northern ; both are on the river Tornea, and nearly on the same meridian. The other stations were mountains not tar from the river. The base which was to be measured was chosen about midway between Tornea and Kittis, and the extremi- ties denoted bv signals. M. 2i>; O. S(J. O * All the geodetical angles were observed in the space of about two months, between the beginning of July and the beginning of September, 17-'>(i. The observations were made with a quadrant of two feet radius. M. :}:i, 7i ; 0. i ) 04. ..!>! !). Is."). The next step was to determine the difference of latitude of the extreme points of the arc. The star d Draconis was selected which passed the meridian very near to the xeiiith ; observations of this star were made at Kittis on the 4th, .">th, (ith, Sth, and 10th of October; and at Tornea on the first five days of November. The difference of zenith-distance was found to be ."> 7 ' --V '. .V>. M. 10k The instrument used for determining this difference of xeni distance was a xenit h--ector made by Graham at London: the jn.-trument resembled that u-ed bv llradh-y in the observations which established the aberration of light. M. o.S. A copper tt-lescope-tube of nine feet long formed one radius of the sector; the extent of the arc of the sector was .V'.\. graduated at every 7'^r. At the focus of the telescope were fixed t wo wires at right angles. ARC OF THE MERIDIAN MEASURED IN LAPLAND. 97 The telescope and the arc formed one instrument. A large pyramid of wood 12 feet high served as the support of the instru- ment. M. 38, 94. The instrument could turn freely round a hori- zontal axis ; it was moved by a micrometer screw acting in oppo- sition to a weight. A plumb-line was suspended from the centre of motion, and marked on the graduated arc the angle through which the instrument had been turned. The absolute zenith- distance of a star at a given place was not determined by the French observers, but only the difference of zenith-distance at two given places. 186. The base was measured on the frozen surface of the river Tornea, very nearly in the direction of the stream ; the extremities of the base were on the land. The measurement was begun on December 21st, and occupied a week. Eight rods of fir were employed, each five toises long; the correct length of these rods was determined by the aid of an iron toise which had been care- fully adjusted to the length of the standard toise at Paris. 0. 137. This iron toise is known henceforth in the history of the subject as the Toise du Nord. A similar iron toise had been taken by the Equatorial expedition, which is known as the Toise du Pdrou. Neither Maupertuis nor Outhier records the fact that these two toises were made at the same time and by the same artist, Langlois ; this we learn from La Condamine : see the Paris Hemoires for 1772, Part II. pages 4*82. ..501. 187. The measurers of the base divided themselves into two bands ; each band had four of the fir rods, and measured independ- ently : the length of the base was found to be 7406 toises 5 feet 4 inches by one band, and 7406 toises 5 feet by the other band. After the measurement was finished three of the party verified that no error could have arisen in counting the hundreds, by using a cord 50 toises long over the whole base. 0. 144. The sun scarcely rose above the horizon, but the twilight, the white snow, and the Aurora Borealis supplied enough light for four or five hours work daily. M. 51. 188. -It followed from the length of the base that the length of the arc of the meridian intercepted between the parallels of T. M. A. 7 98 ARC OF THE MERIDIAN MEASURED IN LAPLAND. Tornea and Kittis was 55023^ toises ; and that the length of a degree of the meridian at the Arctic circle was nearly 1000 toises greater than the length calculated according to the Cassinian theory in the book De la Grandeur et de la Figure de la Terre. M. 58. The party then went to Tornea and remained shut up in their chambers in a kind of inaction until March. The difference between their result and that of the Cassinian theory was so great that it astonished them ; and although they considered their operations to be incontestable, yet they resolved to execute some rigorous verifications. M. 63. We read in the Paris Mtmoires for 1737, page 94 of the historical portion: On la tint fort secrette, tant pour se dormer le loisir de la reflexion sur une chose peu attendue, que pour avoir le plaisir d'en apporter a Paris la premiere nouvelle. 189. The angles of the triangles were supposed to admit of no doubt ; these angles had been observed many times by various persons ; and the three angles of every triangle had been observed. The calculations were verified by combining the triangles in a different series; and also by assuming that errors had arisen in measuring the angles, which all tended to make the length greater than it should have been. But the length of the arc of the meridian still remained without any very decided diminution. M. 63.. .65. The measurement of the base was considered to be also above suspicion ; thus there remained only the very important point of the difference in latitude of the extreme stations ; and accordingly this was redetermined. The star a Draconis was now selected; observations of this star on the meridian were made with the zenith-sector at Tornea on the # 17th, 18th, and 19th of March, 1737, and at Kittis on the 4th, 5th, and 6th of April : the differ- ence of zenith-distance was found to be 57' 25". 85. M. 115. The reason given in the Paris MSmoires for 1737, on page 95 of the historical portion, for going over the astronomical part of the work again is that it could be done much more easily than the other parts. 190. The observations for determining the difference of lati- tude required corrections for aberration, for precession, and for ARC OF THE MERIDIAN MEASURED IN LAPLAND. 99 a third inequality which had been recently discovered by Bradley, and which is called nutation. No correction was applied for refraction. M. 125. See Bouguer's Figure de la Terre, page 290. Thus, finally, the amplitude of the arc of the meridian was 57' 26". 93 by the star S Draconis, and 57' 30". 42 by the star a Draconis ; the difference is 3". 49. Maupertuis considered that 0".95 of this difference was owing to an inequality in the gradu- ation of the sector, which was discovered by careful scrutiny. M. 124. Maupertuis took the mean of the two results, 57' 28". 67 for the amplitude ; and from this he deduced that the length of the degree of the meridian which is bisected by the Arctic circle is 57437.9 toises. 191. Important pendulum experiments were made at Pello, which is close to Kittis. The result is that a pendulum which oscillates in a second at Paris will make 59 more oscillations in 24 hours at Pello than at Paris. M. 172. 192. The Academicians endured great hardships during their operations. The severe cold of the winter months must have been anticipated ; and the precautions which the natives had learned from experience would afford some mitigation of this evil. But the most painful period of the survey seems to have been that which was spent among the mountains in observing the geodetical angles : in one instance they remained for ten days on a mountain. M. 21. The exposure to extremes of heat and of cold, the ex- cessive rains, and the want of proper food, all contributed to the sufferings of the party. But the worst torment seems to have been that inflicted by insects. Maupertuis calls them flies, and says they were of different kinds. M. 14, 16, 22. Outhier calls them by various name's ; flies, gnats, midges : thus cousins 55, 57, 58, 59, 63, 64, 74, 82 ; moucherons 64, 65, 75, 79, 82 ; mouches 57, 58, 64. Le Monnier fell very ill. M. 24 ; 0. 75, 79, 81. Ac- cording to Hutton's Mathematical Dictionary the health of Mau- pertuis was permanently impaired by the hardships he underwent. The Academicians left Tornea in June, 1737, and reached Paris in August. 72 UNIVERC'TY OF CALIFORNIA HFPARTMFNT OS? DUVCIOC 100 ARC OF THE MERIDIAN MEASURED IN LAPLAND. 193. The measurement of the arc of the meridian by the French in Lapland is historically the most important of all such operations. The question as to the oblate or oblong form of the Earth was decisively settled. Two generations of the best astronomical observers formed in the school of the Cassinis had struggled in vain against the authority and the reasoning of Newton. 194. Some incidental matters may now be noticed which present themselves in studying the narratives. Maupertuis says on his page xii. : Sur des routes de 100 degres en Longitude, on commettroit des erreurs de plus de 2 degres, si naviguant sur le Spheroi'de de M. Newton, on se croyoit sur celui du Livre de la Grandeur et Figure de la Terre. I cannot understand this. Nothing is said about the latitude ; but the amount of error in a course of 100 degrees of longitude will depend mainly on the latitude. In the life of Maupertuis in the Biograpliie Universelle, which is partly by Delambre, reference is made to the exaggerations of Maupertuis on this point. Clairaut is the mode of spelling which the bearer of this dis- tinguished name himself adopted: Outhier, however, generally uses Clairaux ; once he has Clairault. 0. 25. Maupertuis, in returning to France, was shipwrecked in the Gulf of Bothnia ; he merely alludes to this misfortune himself : but we find from Outhier that the instruments were immersed, and were cleaned rather more than a month after the accident. M. 78; 0.169,189. 195. The success of the Arctic expedition may be fairly ascribed in great measure to the skill and energy of Maupertuis : and his fame was widely celebrated. The engravings of the period represent him in the costume of a Lapland Hercules, having a fur cap over his eyes; with one hand he holds a club, and with the other he compresses a terrestrial globe. Voltaire, then his friend, congratulated him warmly for having " aplati les poles et les Cassini." See articles .entitled Histoire des Sciences in the Revue ARC OF THE MERIDIAN MEASURED IN LAPLAND. 101 des deux Mondes, Jan. and Nov., 18G9. Readers of Carlyle's History of Frederick the Great will remember the allusions to the Earth-flattener. 196. Although the measurement of the Lapland arc settled the question as to the oblate or oblong form of the Earth, yet it introduced a great difficulty ; for by comparing the result with that obtained from the French arc the ellipticity of the Earth appeared to be about =- . This was greater than had been ex- pected, and greater than subsequent operations, such as that in Peru, furnished. From our present knowledge it is certain that this value of the ellipticity is far too large. We have seen indeed, in Art. 177, that Clairaut assigned as the ellipticity furnished by the Lapland arc ; this must have been obtained by using for the French arc a certain value obtained by Maupertuis in his Figure de la Terre, page 126 ; but this value of the French arc was soon afterwards found to be too small. 197. According to La Lande, Maupertuis himself was not satisfied with his operations. We read in the Bibliographie Astronomique : .... je sais que Maupsrtuis n'en etait pas lui-meme tres-content. Page 407. ... Au reste, on m'ecrit de Suede que Maupertuis s'etait propose de recommencer la mesure a ses depens j ce qui prouve qu'il n'en etait pas t res-content. Page 811. It is well known that the Lapland arc was remeasured at the beginning of the present century by Svanberg and others under the direction of the Stockholm Academy of Sciences. La Lande alludes to the early stages of this operation ; see the Bibliographie Astronomique, pages 811, 837, 857. Svanberg obtained a de- cidedly shorter length for a degree of the meridian than that of Maupertuis, namely, 57196.159 toises instead of 57437.9 toises ; but the middle points of the two degrees are not quite identical. 198. We may just notice the memoir by Celsius, which is contained in the German translation of Maupertuis's Figure de la 102 ARC OF THE MERIDIAN MEASURED IN LAPLAND. Terrc : see Art. 1S1. This is probably a translation of one which was originally published at Vpsal in 17->S under tin 1 title of De observationibus pro riaurd tdlaris detenninandd in tiallid habitis disquisitlo, according to La Lande's Bibliographic Astronotirique, page 4()(j. In the translation Celsius first defends the astronomical opera- tions in Lapland from an objection which had been urged against them by J. Cassini before the Paris Academy, because the sector had not been reversed at each place of observation. Celsius main- tains that this was unnecessary for the purpose of the observers, especially considering the excellence of Graham's sector. Then Celsius proceeds to criticise the French operations recorded in the work De la Grandeur et de In Fiynre de hi Tcrrc ; and ho considers that he shews both the astronomical and geodetical parts to be untrustworthy. These operations indeed were just about to be given up and replaced by the more accurate determinations re- corded in the work La Meridienne de Paris verifv'c. 199. For further information respecting the Lapland arc of the meridian, I may refer to my memoir OH the subject published in the Cambridge Philosophical Transactions, Vol. xn. ; I have there corrected the numerous and serious errors which have been made by distinguished astronomers in their account of this remark- able measurement. CHAPTER VIII. MISCELLANEOUS INVESTIGATIONS BETWEEN THE YEARS 1721 AND 1740. 200. WE have first to consider a production to which allusion has been made in Arts. 143 and 150. It is entitled A Disser- tation concerning the Figure of the Earth, by the Reverend John Theophilus Desaguliers, L.L.D. F.R.S. This is contained in Vol. xxxin. of the Philosophical Transactions: the volume is for 1724, 1725 ; and is dated 1726. The dissertation consists of four parts. 201. The first part occupies pages 201... 222 of the volume. This part criticises the conclusions at which J. Cassini had arrived as to the form of the Earth in his De la Grandeur et de la Figure de la Terre, of which we have given an account in Arts. 100... 108. Desaguliers endeavours to shew that the Cassinian figure is impossible, because it would lead to a deviation of the plumb-line, from the direction which is at right angles to the surface of water, to the amount of five minutes : but the process is unsound. We know now that under certain hypotheses as to the form of the solid nucleus, the outer surface of the fluid might be an ob- longum : see Clairaut's Figure de la Terre, page 224. Desaguliers maintains that the latitudes in the French survey of the meridian cannot be relied on as sufficiently accurate to establish the oblong figure of the Earth ; and he is not satisfied that the heights of the mountains were properly determined. Desagulier's criticisms have perhaps some foundation; but like many controversialists he seems disposed to be unfair. For in- 104 MISCELLANEOUS INVESTIGATIONS FP.OM 1721 TO 1740. stance, he considers that the height of one mountain was over- estimated, and the height of another under-estimated; and thus, he says, we mu>t add 20 toises to the length of the 44th degree of latitude, and take away .'}() toises from the length of the 4.">th degree of latitude. But even admitting these corrections to be necessary, they tend to balance each other; and they produce no perceptible effect on the definite result obtained by Cassini, namely, that the whole southern arc from Paris to the Pyrenees gives a longer average length of a degree than the whole northern arc from Paris to Dunkirk. Strictly speaking, what Desaguliers calls the 44th degree of latitude should be the 45th ; and what he calls the 45th should be the 40th. Desaguliers assigns one reason which may have induced Cas- sini to make the Earth oblong, in these words : <: especially be- cause in this Hypothesis, the Degrees differ most in Length from one another about the 45th Degree." .But this is quite unsatis- factory. For if we suppose the Earth to be nearly spherical, then whether it be oblate or oblong the degrees wil 1 differ most in length at about the 45th degree : see Art. 140. 202. The second part of the dissertation occupies pages 2o!)...255 of the- volume. The object of this part is to shew "How the Figure of the Earth is deduc'd from the Laws of Gravity and Centrifugal Force." Instead of giving anything of his own, Desaguliers transcribes a long extract from K. fill's book a/ainst Burnot ; the extract consists of that matter which Keill took substantially from Huygons: see Art. 74. Desaguliers says : I own indeed that he has made a Mistake in that l>ook concerning the Measure of the 1 Wrees of an Ellipse ; but I find that all that r-latesto the oblate S| >her< .iilicul Figure of the Karth is right.... Tin: nii>iak' "<' course is that which we have noticed in Art. 7<. Desanuliers would probably have thought it unnecessary to warrant the accuracy of the matter which he transcribed, if he had known that it was substantially all due f<> Huygens. MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 105 203. The third part of the dissertation occupies pages 277. . .304 of the volume. This part is chiefly a criticism of the me- moir by Mairan which we have examined in Arts. 109. . .1 14. Much of what Desaguliers says, though quite true, would have failed to produce any effect on Mairan. For instance, according to Mairan, Paris is more distant from the centre of the Earth than a place at the equator is ; hence the attraction at Paris will be less than it is at the equator; hence, although the centrifugal force at the equator is greater than at Paris, we may have gravity at Paris less than gravity at the equator : and this is contrary to observation. But Mairan would have declined to admit the statement in Italics ; he had invented a law of attraction for himself which made the attraction greater at Paris than at the equator. Of course the assailable part of Mairan 's memoir was the arbitrary law of attraction which he had invented ; and against this Desaguliers directs a decisive argument. He finds that, taking Mairan's law, and allowing for centrifugal force, the Paris seconds pendulum would have to be shortened at the equator nearly an inch. He says: "But this being about five Times more than agrees with Observation ; what proves too much, proves nothing at all." See Art, 52. Desaguliers finds, that on Mairan's law the polar and equa- torial columns of fluid would not balance ; but Mairan might have replied that the Earth was solid, and for this reason he might have declined to admit the principle of balancing columns. 204. Desaguliers in the third part of his dissertation returns to the subject of the French arc. He arranges a table which gives the observed latitudes of successive stations on the meri- dian, and also the distance from Paris in toises. He shews that there is not a constant decrease in the length of a degree in passing from the southern extremity of the arc to the northern. But the objection is of no value; because the French observers did not require, and did not attempt to find, the latitudes of intermediate stations with the same accuracy as the latitudes of Paris and of the two extremities of the arc. Desaguliers says on page 303 : 106 MISCELLANEOUS INVESTIGATIONS FROM 17- L TO 174-0. To conclude, I will propose a Method of observing the Figure of the Shadow of the Earth in Lunar Eclipses, whereby the Difference be- tween the Diameters in the oblong spheroidical Figure, if there be such an one as Mons. Cttattt/ti affirms (viz. of % to D5), may be disco ver'd. But the method has, I believe, no practical value. 205. The fourth part of the dissertation occupies pages 34-4, o-i-') of the volume. It consists of an account of an experiment to "illustrate" what had been said in the preceding parts. The essence of the experiment may be thus described. Take a hoop of thin elastic steel ; let it revolve round a diameter as axis, the axis passing freely through the steel : then the greater the an- gular velocity the more will the boo]) bulge out into an oblate form. The toy with which Desagnliers amused himself of course proved nothing to the point ; however, lie boldly asserts that from this experiment, compared with what had been said, "it will appear that the Earth cannot preserve its Figure, unless it be an oblate Spheroid." 2(MJ. There are some incidental matters of interest in the dissertation which may be noticed. Desaguliers suggests on page 20!), that a Degree of Latitude shou'd be measurd at the ^Equator, and a Degree of Longitude likewise measur'd there ; and a Degree very northerly, as for Kxample, a whole Degree might be actually measur'd upon the Jhdtick Sea, when frozen, in the Latitude of sixty Degrees. We read on pages 211), 220: when once an Hypothesis is set on Foot, we arc too apt to draw in Circumstances to coniirm it; tho', perhaps, when examin'd im- partially, they may rather weaken, than strengthen our Hypothesis;, otherwise, the Author of the History of the Hntj-d A ,,,,,',>,,/, for the Year 1713, wou'd not have alledufd, that tic 1 laic. M<n<x. ( 'assini obiwrrd Jupiter t<> b't urn!, a.s a I 'roof of young Mmis. Caxsiiii* Hypothesis; be- cause Ji'int'-i' is oval the other Way, that is, an oblate Spheroid flatted at the roles... 13ut. I cannot find anything in the volume which justifies this remark by Desaguliers. MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 107 The only reference to Jupiter occurs after a notice of the fact that the Earth deviates but little from a sphere; then we read : Si Jupiter est ovale, comme il 1'a paru quelquefois a feu M. Cassini, il faut qu'il le soit bien davantage pour le par6itre de si loin. It is obvious that these words do not bear any such meaning as Desaguliers suggests. Desaguliers refers to the opinion of Dr Burnet, which we have noticed in Art. 74. Desaguliers says on his page 221 : "But Dr. Burnet, afterwards, alter'd his Opinion, as I am credibly inform'd." Desaguliers asserts "That a fluid Substance, of any Figure, will by the Gravity of its Parts become spherical, ..." He gives what he calls a demonstration of this on his pages 278, 279 ; but, as might be expected, his demonstration is quite inconclusive. See Art. 130. Desaguliers adopts on his page 280 the erroneous notion that by increasing the density of the central part qf the Earth, the ellipticity is also increased ; see Arts. 30, 84 and 172. Newton and David Gregory do not state whether they suppose the central part still to remain fluid or to become solid. Desaguliers, how- ever, says distinctly, " Then if, when the Central Parts are fix'd, and the superficial Strata are still fluid, ..." To shew that Desaguliers is wrong, we have only to put a = on page 219 of Clairaut's Figure de la Terre ; then we find that K I S is less than ~ . Or see Simpson's Mathematical Dissertations, page 30. A paragraph which occurs on pages 280 and 281 is to be cancelled, according to an Advertisement by Desaguliers at the end of Number 399 of the Philosophical Transactions. 207. Desaguliers, on his page 285, deviates from accuracy in saying that " on different Parts of the Surface of the Earth (in the Condition it is now) the Gravity on Bodies is reciprocally as their Distance from the Centre of the Earth." I have already stated that this proposition should be enunciated thus : Gravity 108 MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. resolved along the radius-vector varies inversely as the radius ; see Art. 33. Desaguliers omits the resolution along the radius-vector. Moreover, I think from his context, and from a calculation on his page 287, that he made another mistake, and supposed that the attraction along the radius-vector varied inversely as the radius ; that is, I think, he neglected the distinction between attraction and gravity. On his pages 286 and 287 he assumes that for an oblongum the gravity will vary inversely as the radius-vector; and by gravity he means here attraction alone, for he proceeds to allow separately for the centrifugal force. The assumption is unjustifiable, and seems to have arisen from the confusion of gravity with attraction in the case of the oblatum. 208. Desaguliers obtained from a friend a " Philosophical Argument" against Mairan ; it is thus stated on his page 298 : If the Earth was of an oblong spheroidical Figure, higher at the Poles than the -^Equator ; the Axis of its Revolution, wou'd either go thro' one of its short Diameters, or be continually changing unless the said Axis did exactly coincide with the Axis of the Figure. These words themselves are true ; they are, however, appli- cable to the oblatum if we change short into long. The so-called demonstration which follows shews that Desaguliers and his friend were wrong in their notions on the subject. In modern language these notions amount to considering that the rotation of an oblongum round its axis of figure is unstable- The mechanical knowledge of the period was inadequate to the discussion of a difficult problem in Rigid Dynamics. 209. A work was published at Padua in 1728, entitled Joannis Poleni...Epistolarum Mathematicarum Fasciculus. The work is in quarto ; the pages are not numbered. One of the letters relates to the Figure of the Earth ; it is addressed " Viro celeberrimo Abbati Gui. Grando." This letter occupies eleven pages ; it is of little importance. Since some persons maintained that the Earth was oblate, and others that it was oblong, Poleni considers it safer to adopt the spherical form as a compromise between the two extremes. He MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 109 suggests, however, that by measuring an arc of longitude, say in latitude 48, a test might be obtained as to the two extreme hypotheses. For, assuming the same perimeter of the meridian in the two cases, the arc of longitude would be much shorter if the figure be an oblongum than if it be an oblatum. Poleni states that for an arc of one degree of longitude, the difference would be about 777 toises. See Art. 215. He considers that the spherical form may be reconciled with the existence of centrifugal force, by supposing the Earth not to be homogeneous. 210. Some pendulum observations were made^at Archangel in 1728 by L. Delisle de la Croyere. They are recorded in the Commentarii Academice...Petropolitance,Vol. iv. which is for 1729, and was published in 1735 : see pages 322... 328 of the volume. 211. In the Paris Mdmoires for 1732, published in 1735, there is a memoir entitled Reponse aux Retnarques qui out ete" faites dans le Journal Historique de la Rtfpublique des Lettres sur le Traittf De la Grandeur et de la Figure de la Terre. Par M. Cassini. The memoir occupies pages 497... 513 of the volume. In the Journal Historique de la Rtpublique des Lettres for January and February, 1733, some extracts were given from several printed letters of the Marquis Poleni ; among these letters one related to the Figure of the Earth : see Art. 209. The editor of the Journal added some remarks impugning the accuracy of the observations and the soundness of the results given in the work De la Grandeur et de la Figure de la Terre. J. Cassini replies to the remarks. The chief point urged in the remarks seems to be that some of the observations of latitudes recorded in the work differ con- siderably from the latitudes finally adopted ; the chief point urged in the reply seems to be that observations made with less care and with small instruments were rejected in favour of observa- tions made with more care and with larger instruments. The reply seems to me temperate and able. There is on pages 512, 513 a list of the misprints which had been detected in the work De la Grandeur et de la Figure de la Terre. 110 MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. The following succinct account of the French survey of the meridian is given on page 4'KS : Cet ouvrage f'ut propose par mon Pore, et prolonge en 1 (84 jusqu'au dela tie Bourses vcrs le Midi, pendant quo M. de la Hire y travailloit du cute du Xord. Jo 1'ai continue avec nion Tore et M. Maraldi, depuis Bourges jusqu' a Collioure en 17<>0 ot 1701, et a pros 1'avoir acheve entiereineiit en 17 IS avee M rs . de la Hire le ills et Maraldi, en le prolongeant jusqu'a. I'extremite septentrionale du lloyaume, j'en ai donne le re.sultat au Public ; ainsi e'e.st a nioi a en proud re la defense. 212. In the Paris Me moires for 1733, published in 1735, there are five memoirs which are connected more or less closely with our subject. A connected account of them is given in pages 46... 03 of the historical portion of the volume. The first memoir is by Maupertuis; we have noticed it in Art. 131. 213. The next memoir is entitled Mrthode pratique de tracer sur Terre un Parallels par un degre de latitude donne ; et du rapport da meme Parallels dans le Spltero'ide ollony, et dans le Sphe'ro'tde applati. Par M. Godin, The memoir occupies pages 223... 232 of the volume. The memoir shews that for various reasons the accurate deter- mination of the latitude of a place is not an easy problem in practical astronomy. Nevertheless it is maintained that an arc of longitude may be traced without much difficulty ; and the best way of conducting the operation is explained. Some numerical results are given as to the length of a degree of longitude ; and remarks are made on the letter of Poleni which we have noticed in Art. 209. (Indin finishes witli determining the arcs common to an oblaturn and an oblmigum which have the same centre, and their axes in the same straight line. Tim matter is very simple, but the account which is given of it in page 53 of the historical portion of the volume is not altogether intelligible. 214. Tin; next of these memoirs is entitled Description d'un Instrument fjm j>ei/t xemr d determiner, .stir la surface de la Terre, toufi les points d'un ('crclv />araf/cle a I Eqiiateur. Par M. J)c La, Condamine. The memoir occupies pages 2!)4...3()1 of the volume. MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. Ill The instrument is intended to facilitate the operation described in Godin's memoir ; but it does not seem to me that it would be of any practical value. An extract of a letter written from Quito by La Condamine is given in the volume of Mgmoires for 1734, which shews that he had himself discovered grave faults in the memoir, and requested that it might not be printed. 215. The next of these memoirs is entitled De la Carte de la France, et de la Perpendiculaire ob la Mtridienne de Paris. Par M. Cassini. The memoir occupies pages 389... 405 of the volume. The memoir gives an interesting account of the operations in tracing a line perpendicular to the meridian of Paris westwards to the coast of Normandy. Cassini finds that the length of a degree of longitude in the parallel of St Malo is 36670 toises ; and he says that on the supposition of the spherical form of the Earth it should be 37707 toises. Hence he infers that the Earth must be of an oblong form. It will be observed that the discrepancy here is very wide ; and a less extravagant result was obtained by Cassini in the M&noires for 1734 : see Art. 220. Results much more moderate than this were obtained by Cassini de Thury in the Me'moires for 1735 and 1736 : see Arts. 224 and 226. It will be convenient to place here the formulas relating to this matter. Let X denote the latitude, p the corresponding radius of curva- ture of the meridian, r the radius of the section parallel to the equator. If the earth were spherical, we should have r = p cosX. If the earth is an oblatum, a denoting the semi-axis major, and e the excentricity of the generating ellipse, we have a (1 - e 2 ) acosX and r = r~^r Thus it is obvious that r is now greater than p cos X 112 MISCELLANEOUS INVESTIGATIONS FKOM 1721 TO 1740. If therefore it appeared by observation and measurement that r is less than p cos X, it would follow that the Earth could not be an oblatum. The values of p and r in the case of the oblatum are often required in our subject. 216. It was found that the distances between places deter- mined by the trigonometrical operations in France were in many cases less than had been previously supposed ; and Cassini makes the following obvious remark : ...... ce qui vient apparemment des grands detours quel'on est oblige" de faire pour chercher des routes praticables, joint a ce que les mauvais chemins paroissent toujours plus longs qu'ils ne le sont reellement. The operations terminated at Bayeux; Cassini says, after speaking of St Malo : Nous allames de-la a Bayeux ou nous fimes diverses observations de hauteurs du Soleil, d'Etoiles fixes, et principalement de 1'Etoile polaire, dans le Palais Episcopal qui joint & la Cath6drale, et ou M. l'Evque de Bayeux a fait tracer dans sa bibliotheque une grande Meridienne, avcc des lignes qui marquent les heures avant et apres midi, de cinq en cinq minutes, par M. TAbbe" Outhier qui a travaille" avec nous a la descrip- tion de la Perpendiculaire depuis Caen jusqu'a St Malo. The last of the five memoirs is by Clairaut ; we have noticed it in Art. 160. 217. We have a memoir on pendulum observations in pages 302... 314 of Number 432 of the Philosophical Transactions. The Number is for the months of April, May, and June, 1734, and forms part of Vol. xxxvin. which is for the years 1733, 1734, and is dated 1735. The memoir is entitled An Account of some Observations made in London, by Mr. George Graham, F.R.S. and at Black- River in Jamaica, by Colin Campbell, Esq.; F.E.8. con- cerning the Going of a Clock ; in order to determine the Difference between the Lengths of Isochronal Pendulums in tlwse Places. Communicated by J. Bradley, M.A. Astr. Prof. Savill. Oxon. F.R.S. The observations were made during 10 days in England, and during 26 days in Jamaica. Bradley deduced from them that the seconds pendulum of London lost 1 minute 58 seconds in a MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 113 day at Jamaica ; and from this he obtained for the ellipticity of the Earth the value - . Bradley gives the reasons which led him to "esteem Mr. Campbells Experiment to be the most accurate of all that have hitherto been made..." This memoir is referred to by Stirling in the Philosophical Transactions, Vol. xxxix. page 103 ; by Clairaut in the Philo- sophical Transactions, Vol. XL. page 291 ; and by Maclaurin in his Fluxions, Art. 6G4. 218. In the Paris Mtmoires for 1734, published in 1736, there are four memoirs which are connected more or less closely with our subject. The first of these memoirs is entitled Methode de verifier la Figure de la Terre par Parallaxes de la Lune. Par M. Manfred*. The memoir occupies pages 1...20 of the volume; there is an account of it on pages 59... 63 of the historical portion of the volume. Supposing the Earth not to be spherical, the parallax of the Moon will be different at different places on the Earth's surface, even when all other circumstances are alike. Manfred! suggests that observations of the Moon taken at two distant places, nearly on the same meridian, would therefore supply in- formation as to the figure of the Earth. In spite of the errors to which such observations might be liable, he maintains that it would be possible to decide in this way the question as to the oblate or oblong form of the Earth, 219. The next of these memoirs is entitled Comparaison des deux Loix que la Terre et les autres Planetes doivent observer dans la figure quc la pesanteur leur fait prendre. Par M. Bouguer. The memoir occupies pages 21... 40 of the volume ; there is ah account of it on pages 83... 87 of the historical portion of the volume. This memoir is important in the history of Hydrostatics. The two principles to which it refers, are Newton's principle of balancing columns and Huy gens' s principle of the plumb-line. Bouguer 's, T. M. A. 8 Ill MisCT.T.LANKors INYKSTK.'ATIOXS FROM 1721 TO 1710. object is t slu-w that under certain conceivable laws of force either principle might he satisfied, while the other was not; ami then there could not he equilibrium. The whole matter is now well undeistood : and it is admitted that tor equilibrium the forces acting must >ati>iy a certain condition, namely, in ordinary nota- tion, supposing the thud homogeneous., AV.r -f Y<lij -f Zdz must be a perfect differential ; and it is known that this condition is satis- tied tor such forces as occur in nature. Boug'Uer savs on Ins tirst page: Kutiv plusicurs Mathematicians d'uu grand nom qni onttotirne leiir vuc vers cette mature, M. Ilugwns et M. Herman sont les souls qui out .applique en meine temps les deux luix ; ils out trouve quVlles s'aeconloieiit u domier a la Xerre iiue uieme figure dans les suppositions particiilieres d'uue jicsauteur originaircment coustaiite. et d'uue pe- santeur proportionnelle aux distances an centre. This statement is correct with resjiect to Iferniann ; but there seems no authority for it with respect to Huyyeiis. Hermann did consider both principles and both the laws of attraction: see Arts. !)! and [)">. Hny^ens contined himself to the nsc> of Newton's principle, and to the supposition of a constant attraction: sec Arts, ."land .V>. In his investigations, Bouguer, as we should now say, considered only forces in one plane. He supposes the direction of the force to be always perpendicular to a given curve. This hypothesis was afterwards discussed by Clairaut in pages (>'>... 77 of his Figure de la Tvrrc. Clairaut shews that, in order to render this hypothesis reasonable, 'wo must suppose a solid nucleus to the fluid : see his page< (5! and 71. Although Boiiguor's own examples are not of e-Teat value, U -cause they depend on laws of force which can liardlv be con- sideivd natural, \'et the memoir must have been \cr\- useful at the iini'-. as it called attention to an important subject, and pro- bahly suggested to ( 'lairaut the occasion of his own investigations. --'*. The next of these memoii's is l>v Maupertuis ; we have noticed it in Arts. I '.\'l. . . 1 M!>. I li ( ' last ot tlie.-e memoirs i> entitled Ih'l'i Perpend ic uldire (i ! .!/' I-I'/K mil- <1<: /'art*. >/'uln/if>'c rc/'.y / (>ricitf. 7'c/r J/. ('(tssiitt. MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 115 It occupies pages 434... 452 of the volume; there is an account of it on pages 74... 77 of the historical portion of the volume. This memoir contains an account of the operations in tracing a line perpendicular to the meridian of Paris, eastwards to Stras- bourg ; the operations and the memoir are in continuation of those which we have already noticed : see Art. 215. Cassini finds that the length of a degree of longitude in the latitude of Strasbourg is 37066 toises ; and he says that on the supposition of the spherical form of the Earth the length would be 37745 toises. Hence he infers, as before, that the form of the Earth must be oblong. The result differs very considerably from that given in the M&noires for 1733 : see Art. 215. The present result depends of course on the longitude of Strasbourg ; and this is determined by the aid of observations formerly made by Eisenschmidt. Cassini assumes credit to himself for taking a mean between three determinations, though less favourable to his theory of an oblong form than the value which Eisenschmidt him- self adopted. Thus we read at the close of the account in the historical portion of the volume, with respect to these observations : mais enfin ces observations se sont trouvees si favorables au Spheroi'de allonge, que M. Cassini a eu la moderation de n'eu pas vou- loir tirer tout 1'avantage qu'il cut pu a la rigueur, et de s'en retrancher une partie. 221. A double prize was offered by the Paris Academy for the year 1734 ; the subject related to the inclination of the planes of the orbits of the planets to the plane of the Sun's equator. The prize was divided between John Bernoulli and his son Daniel. The essay by Daniel Bernoulli is memorable in the history of the Mathematical Theory of Probability : see my History, page 223. The essay by John Bernoulli is reprinted in his Opera Omnia, Vol. in. pages 261... 364, under the title Essai dune nouvelle Physique Ce'leste.... Pages 345... 35 5 relate to the Figure of the Earth ; but it would be a waste of time to discuss them. The essay uses a system of vortices ; and as those who invented such visionary machinery were guided by no principle and restrained by no law, they could easily arrive at any result they pleased. 82 116 MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 17^0. Jolin Bernoulli disliked and depreciated Newton, and he was now competing for a prize from tin- Paris Academy; he had, therefore, a d<.ulih' iva>on for taking tin- side of error. This he dors much to his own satisfaction, and concludes thus in the lan- guage of premature triumph : A pros cette heureuso conformito do notrc theorie, avoc los observa- tions celestes. pent-oil plus long-temps refuser u la Terre la figure do splieroide oblong, foiide d'ailleurs sur la dimension des d ogres de la meridieime, ontreprise et exeeutee par lo ineme 31. Cassini, avec line exactitude inconcevable ? 222. In the Paris Mf moires for l7^-">, published in 173$, we have some memoirs which bear, though slightly, on our subject. An account of them is given on pages 47... b'.") of tin- historical portion of the volume; but the last six pathos of this account refer to some memoir attributed to ( 'lairaut. which does not seem to have been published. According to this account, an arc of longi- tude, if measured in a very high latitude, might be expected to yield as good a result as an arc of meridian. Douguer, however, in an able memoir published in the volume for 17o(>, shewed that this expectation was (piite unfounded. The first memoir is entitled Mt'thode de determiner si la Terre c*t 8j)1i('ri(/n ou iinn, ct Ic rapport dc *cs dcyn's entreu;r, taut stir /f.v Mf'ridi<.'nH que sur TJ^uatcmr ct xcs Paralleles. J'ar J/. Cassini. The memoir occupies pages 71. ..86. The idea of the memoir can be easily stated. Select a mountain, from which the sea is visible in various directions, and observe the dip of the horizon. If the Karth is spherical, the dip will he the same in all directions. If the Karth is not. spherical, the dij> will be different in different directions. J5y observing the dip in tin- directions t the meridian and of the prime vertical, ('as-ini shews that a sensible diiVerencc oiivht to be obtained on the two current hvpotheses as to the form of the Karth ; and that thus the (juestion between the two hypotheses might be settled. I pre-uni' 1 , however, that the method has never been found of an v use in pract ice. MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 117 The next memoir is by Maupertuis ; we have noticed it in Art. 140. The next to this is by Clairaut; we have noticed it in Art. 161. 223. The next memoir is entitled Seconde Methode de dtier- miner si la Terre est Spherique ou non, indtfpendamment des Ob- servations Astronomiques. far M. Cassini The memoir occupies pages 255. ..261 of the volume. The idea of the memoir can be easily stated. Take two points A and B on the same meridian ; say the summits of two mountains. At A observe the angle which AB makes with the vertical at A ; at B observe the angle which BA makes with the vertical at B. Let the verticals at A and B, when produced, meet at 0. Let the distance AB be measured. Then by solving the triangle ABO we can find AO, which may be considered as the radius of curvature at A of the arc AB. Take a third point 0, which is due East or due West of A. Then in the same way we may determine the radius of curvature at A of the arc AC. If the Earth is a sphere, we ought to obtain the same value of the radius of curvature in the two cases ; if the values obtained are different, we have information which may serve to settle whether the form is oblate or oblong. The method is substantially the same as was used by Riccioli in attempting to find the size of the Earth towards the middle of the seventeenth century. See De la Grandeur et de la Figure... pages 29 6... 306. I believe the method is of no practical value. 224. The next memoir is entitled De la Perpendiculaire d la Mtridienne de Paris, dforite a la distance de 60000 Toises de r Observatoire vers le Midi. Par M. De Thury. The memoir occupies pages 403... 413 of the volume. M. De Thury was a son of Jacques Cassini, and is usually called Cassini de Thury. The perpendicular was traced from the meridian of Paris to the western coast of France. Cassini de Thury finds that the length of a degree of longitude in the parallel of Brest is nearly 300 toises shorter than it should be on the supposition of the spherical form of the Earth. Hence he infers that the Earth must be oblong. 118 MISCELLANEOUS 1NVKST1CATK >NS FIUM 1 7lM TO 1740. It must however be observed that tor Nantes, which has nearly the same latitude. ( f assini tic Thury obtained a difference of 7^1 toiM's. It is surprising that such discordant results were con-ideivd to lie worth preserving. It is plain that the obser- vations \\vre not go ><\ enough to furnish trustworthy iniereiices. CasMni de Thury assigns 4~"1.'>'S" for the latitude of Nantes, which agrees with the modern value. But he assigns 47" .13' 2'' for the latitude of Brest; and the modern value is 4 s . '2%' '22". See the table published in the ('o)tnaissance des Temps. There must ufcoiir.se be some error in his figures. "22^. The volume for 1735 contains also some important memoirs on the length of the seconds pendulum. A memoir bv Mairan on pages 153... 220 relates to the length at Paris; there is an account of this on pages 81. .. ( J2 of the historical portion of the volume. A memoir bv CJodin relates to the lengths at Paris and at St Domingo. A memoir bv P>ouguer relates to the length [it St Domingo. A memoir by La Condaminc relates to the length at St Domingo. These three memoirs will be found on pages 505.. .544 of the volume. There is some notice 1 of the memoirs by Godin, Bouguer, and La ( 'ondamine on pages 115. ..117 of the historical portion of the volume for 17o(5. "We are told that these investigators did not arrive in Peru so soon as they had hoped ; and it is added : "Mais quo'nnfils nc pusseiit pas encore s'occuper du principal objet de lour Voyage, la Nature est par-tout, et ils tmuvoient par-tout a ol i>erver." '2-i't. In the Paris 3f{'/ntn'rcx for 17*>li, published in 1731), we have four memoirs bearing on our subject. 'Hi,, first memoir is bv Clairaut ; we have noticed it in ,\ n _ \(\ The next memoir is by Maupertuis ; we have noticed it in Ait. 1 H. Tl,,. third memoir is entitled Km- la PcrpemUciilaire t} la Mcridienne de CObsercatotre <i l<> distance dc (JOOOO loises rers le MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 119 Nord. Par M. Cassini De Thury. The memoir occupies pages 329.. 341 of the volume. There is an account of the memoir in pages 103 and 104 of the historical portion of the volume. The perpendicular was "traced from the meridian of Paris to the western coast of France. According to these operations the length of a degree of longitude in the parallel of Brest is 31'0 toises shorter than it should have been on the supposition of the spherical form of the Earth. Hence, as before, it is inferred that the Earth must be oblong. It seems, from what is stated on pages 332 and 333, that in the operations before the present, the angle subtended between two objects had not been distinguished from the projection of the angle on the plane of the horizon. It was sometimes found necessary to construct scaffolds on the tops of lofty trees ; one tree so used was above 100 feet high. Then we read on page 104 of the historical portion of the volume : " Ces Edifices hardis demandoient que ceux qui s'en servoient, le fussent aussi." 227. The last memoir is entitled De la maniere de determiner la Figure de la Terre par la mesure des degrts de Latitude et de Longitude. Par M. Bouguer. The memoir occupies pages 443... 468 of the volume. Bouguer obtains expressions for the length of a degree of the meridian and for the length of a degree of longitude, assuming the Earth to be an ellipsoid of revolution. Then from the lengths of two different degrees he deduces the ratio of the axes of the Earth. By the aid of the Differential Calculus he finds the change in this ratio produced by a given small change in one of the elements on which it depends. Bouguer makes some interesting remarks on what he calls " la differente delicatesse de la vue des Observateurs," or as we now call it the personal equation of observers, see his page 457. He says that if two astronomers have observed several times together and know what we call their personal equation, yet this may be altered by the fatigues of a voyage, by the changes in the body, or by a greater or less density of the atmosphere. 120 MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. Bouguer's main conclusion is that attention should be given almost exclusively to the measurement of arcs of meridian, since practically arcs of longitude could not be determined with sufficient accuracy to settle the question of the Earth's form. 228. We have next to notice A Proposal for the Measurement of the Earth in Russia, read at a Meeting of the Academy of Sciences of StPetersbourg, Jan. 21. 1737. by Mr Jos. Nic. de L'isle, first Professor of Astronomy, and F.R.S. Translated from the French printed at St Petersburg, 1737. 4fo. By T. 8. M.D. F.R.S. This paper occupies pages 27... 49 of Number 449 of the Philosophical Transactions. The Number is for the months January... June, 1737, and forms part of Vol. XL. which is for the years 1737, 1738, aud is dated 1738. The paper is very interesting; it gives an account of the history of opinion on the Figure of the Earth. The work of Eisen schmidt is cited, and its full title reproduced, which agrees with that in La Lande's Bibliographic Astronomique, page 324 : but here it is added pag. 54. cum fig. The paper was written after the French expeditions had gone to Peru and to Lapland, but before the results of their measure- ments were known ; however, some pendulum observations reported by both expeditions favoured the oblate form. An Extract of a Letter from Delisle is given on pages 50, 51 of Vol. XL. of the Philosophical Transactions ; from this it appears that he measured on the ice a base of 74250 English feet, as the commencement of the proposed operations in Russia. In the work by F. G. W. Struve, entitled Arc du Mtridien de 25 20' entre le Danube et la Mer Glaciale... there is a slight notice of Delisle's project : see Vol. I. page viii. The title of the original document is given thus ; Projet de la mesure de la Terre en Russie. Saint-Petersbourg, 1737, 4to. It is stated that Delisle himself published no account of the measurement of the base or the angles. His manuscripts -were preserved in the Observatory of Paris, and examined in 1844 by M. 0. Struve. Delisle was brother to the person who made the pendulum observations at Archangel in 1728 : see Art. 210. MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 121 229. We have next to consider a memoir by Euler, entitled De attractione corporum sphaeroidico-ellipticorum. This memoir is contained in the Commentarii Academice,.. Petropolitance, Vol. X. which is for 1738; the date of publication is 1747. The memoir occupies pages 102... 115 of the volume. The memoir finds expressions in the form of infinite series for the attraction of an oblatum on a particle at the pole, and on a particle at the equator. In the former case the series is not com- plicated, and converges rapidly; as Euler says vehementer convergit. In the latter case the series is very complicated, and this case of the problem cannot be considered to be really solved. We are not told at what date the memoir was read to the Academy ; so that there may have been merit and value in it at the time ; but before the volume was published the solution of the problem by Maclaurin and by Simpson had appeared, in which the results were expressed in exact finite forms, so that Euler's memoir was completely superseded. I have not verified all the work in this memoir. I will give some indication of Euler's method. Required the attraction of an elliptic lamina on a point directly over the centre of the lamina. Let c denote the distance of the point, Sc the thickness of the lamina. Then the attraction is rc cdxdy where the integration is to extend over the whole area of the a? y* ellipse 2 + JL = 1. Integrate first for y\ thus we find that the attraction is equal to A j * T (a?-x*)*dx 4:bc dc - - - . J o (c 2 + x*) (a 8 (b* + c 2 ) + x* (a 2 -b*)}* By expanding, this becomes 1 1.3, 1 . 3 . 5 6 where JT stands for a j (b* + c 2 ) 12:2 MISCELLANEOUS INVESTIGATIONS FRoM 17-1 TO 1740. This expansion will not give a convergent series throughout the range of integration unless a~ l>~ is less than //"' + <". Enler, however. docs not pay any attention to this point. Moreover, he also expands .. in ascending 1 powers of .*.- before the integra- tion, so that this expansion is really not permissible if a is greater than c. However, Euler evaluates in this way the expression namely, by expanding the denominator, integrating eaeh term separately, and then summing the infinite series which arises. We should now of course avoid the expansion. Bv putting a sin 6 for jc, the expression becomes 77 a-C(**8cl9 , . |^ c 2 + a 2 . 2/1J that is , " + a sm^ J c~ 4- a" snr TT f v 7 Vr + r 2 ) 1 that is 2 J_L_JiJ. Hence in the required attraction we have the terms 27rV(ft 2 -fc% 2wic , '7v ., - OC - //; ., -- OC. ay(b +c~) a\f(b~ + c~) N(,-xt consider the term which arises from ,:~. We may proceed thus without expansion : {' n~ :r" ,'- j :i d.r [(c~ + .f- r~ } : ,, o =-;,--., .. --:, - (a x / </.r. J C 2 -f 2C J I''' + ''" C a -f / j Tlien taking the integrals between the limits and a, we obtain 7T 2 TT<? (Y '(ir -I- c 2 "V" ^ ( - Jlfiiff in tlie i-e(jiiin-il attraction we have the fi-nns _ 1 <r - Ir 4//rSc (TT^/- _ Trr'- 1 A/U/ 2 + c 2 ) "" i ^' J (Ir -f c") ^ v ; & -f c' J ; "( 4 " -1 p MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 123 7r6c 2 (o* - V) V (a 8 + c 2 ) 8c 7r5c 3 (a* - 6 2 ) Sc Trfrc (a 2 - 6 2 ) Sc a s (6 2 +c 2 )* a 3 (6 2 + c 2 )* 2a(6 2 + c 2 )* ' Similarly we might proceed with the term which arises from z 4 , which will introduce (a 2 b*)* ; and so on. The result of course will be very complicated. Euler seems to me to increase the complication by putting and expanding the latter in powers of a 2 Z> 2 . He offers a reason for this which I do not quite comprehend. " Vel cum ad applica- tionem ad computum expediat ipsas series retinere, quo singulorum terminorum integralia algebraice exhiberi queant..." Euler's approximate values for the attraction at the pole /I 4e 2e 2 \ and at the equator are respectively 4vrb (= + I , and \O lo 2iL J 4-7T& ( Q + - I ] , where b is the polar semi-axis, and b(l + e) \O O OO / is the equatorial semi-axis. It will be found on examination that these are correct : see Art. 153. Euler applies his results to determine the ratio of the axes in order that a rotating fluid oblatum may be in relative equilibrium ; he obtains a value for the ellipticity, which is sensibly the same as Newton's in the case of the Earth. 230. A few words may be given to the treatise published by Daniel Bernoulli at Strasbourg in 1738 under the title of Hydro- dynamica, although it is rather beyond our subject. On pages 244 and 245 Daniel Bernoulli solves the problem of determining the form for relative equilibrium of the free surface when fluid in a cylinder rotates round a vertical axis ; the angular velocity is not assumed to be the same throughout the mass. The solution is correct, and is recognised as such by Clairaut in his Figure de la Terre, page 55. Daniel Bernoulli however proceeds on page 246 to make some unsatisfactory remarks on vortices. He begins by saying that he thinks the fluid cannot continue permanently in its state if the centrifugal force increases from the axis to the circumference : the context seems to shew that instead of increases he meant de- 124 MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. creases. But it is plain from his remarks that the subject was not understood at the time. Daniel Bernoulli criticises implicitly Propositions 51 and 52 of Book ii. of the Principia, which he considers do not both correspond to possible cases. 231. The volume of the Paris Memoires for 1739 was pub- lished in 1741. On page 30 of the historical portion there is a short notice of a memoir communicated to the Academy by D'Alembert. The memoir does not bear on our subject, but it is interesting to observe the early appearance of a writer with whom we shall be much occupied hereafter. We are told that : " On a trouvd dans M. d'Alembert beaucoup de capacite et d' exactitude." The later writings of D'Alembert do not in general seem to me to deserve the praise of exactness. A memoir by Clairaut occurs in the volume ; of this we have given an account in Art. 177. There is a memoir entitled Sur les Operations Gtomtiriques faites en France dans les anne'es 1737 et 1738. Par M. CassiniDe Thury. The memoir occupies pages 119... 134 of the volume. The operations were chiefly directed to surveying parts of the coast of France, with the view of rectifying the maps. Some observations as to the velocity of sound are recorded. 232. The Academy of Sciences at Paris proposed The Tides as the subject for a prize essay in 1740. Four essays were published in consequence at Paris. One essay was by a Jesuit named Caval- lieri ; this adopted the Cartesian system of vortices. The other essays were by Daniel Bernoulli, Maclaurin, and Euler ; these are reprinted in the Jesuits' edition of the Principia, and it is stated that many errors in the original impression have been corrected. I have used the reprint in consulting these Essays. It will be convenient to postpone an account of Maclaurin's essay until we have examined the part of his Treatise of Fluxions which relates to our-subject; for this contains all that was in the essay with great additions and improvements. 233. The second chapter of Daniel Bernoulli's essay contains some lemmas relating to the Attraction of Bodies. The result MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 125 may be summed up thus : he determines the attraction at any superficial or internal point of an ellipsoid of revolution which is nearly spherical, neglecting powers of the ellipticity beyond the first. The method used consists in finding accurately the attrac- tion of a sphere, and then approximately the attraction of the difference between the sphere and the ellipsoid on a particle at the pole or at the equator; as we have stated in Art. 165 this method had been previously used by Clairaut. But Daniel Ber- noulli seems to claim the method as his own ; he says at the end of his second Chapter : Ceux qni voudront employer 1'analyse pure pour la solution de DOS deux derniers Problemes, se plongeront dans des calculs extremement penibles, et verront par la 1'avantage de notre methode. Although Daniel Bernoulli employed attraction for the purpose of his essay, yet he seems to have had but a weak faith in the principle : see his Chap. I., Art. 6, and his Chap. II., Art. 1. Daniel Bernoulli added nothing to our subject ; all his results respecting Attraction are included in the formulae given by Clairaut in 1737. But his theory of the Tides is very important in the history of that subject, though it would be out of place for us to discuss it here. An account of Daniel Bernoulli's essay was published in 1830 by the late Sir J. W. Lubbock ; it is in octavo, entitled Account of the " Traite 9 sur le Flux et Reflux de la Mer " of Daniel Bernoulli ; and a Treatise on the Attraction of Ellipsoids, pages vii. + 47. 234. Euler's essay on the Tides contains scarcely anything that concerns us. He finds the attraction of a spherical shell on an internal particle in his Art. 20. The results in his Art. 30 are interesting as examples: we will state them. The attraction of the Sun, or of the Moon, at the surface of the Earth, is of course not strictly the same as the attraction at the centre ; hence arises a disturbing attraction as it may be called, which at a given place will depend on the zenith-distance of the attracting body. Euler finds that the number of oscillations made by a pendulum when the Sun and the Moon are together in the zenith is to the number made in the same time by the same pendulum when the Sun l'2(] MisrKLLANKors INVESTIGATIONS FR >M 17-1 TO 1740. and tin 1 Mm ni are together in the horizon as 4(~>()(>M>6 is to 4(i()()(i(i7. Also if the Sun and the Moon are together at 45" from the zenith, first on one side and then on the other side, in the same great circle, the plumb-line on the whole experiences a deviation of less than of a second. These results arc obtained of Course by using the values then adopted for the masses and the distances of the Sun and the Moon. The following passage occurs at the beginning of Eider's Article 1'2: Explosis hoc saltern tempore qualitatihus oceultis missaquo Anglo- ruin quorum dam renovata attractione ...... At tirst sight this looks as if Kuler intended to reject the principle of attraction ; but we find on examination that he prac- ticallv adopts the principle, alter assuming the existence of a subtle thud in order to account for it to his own satisfaction. 23.">. A work entitled .Ay//v' <lu Mi'rulicn entrc I\tris et Amiens... WAS published in 1740. I have not seen the original but only a (lennan translation published at Zurich in 174:2: 1 mu>t assume therefore that the translation corresponds to the original. Mauportuis and his companions in the polar expedition were charged with the business of verifying the length of a degree of the meridian assigned by Picard. They assumed the accuracy of Picard's terrestrial measurement, but determined the amplitude of the arc afresh. The observations were made in the latter half of the year 17'W; the instrument employed was the same zenith-sector as had been employed in Lapland. The l)i n.k' contains a description of the sector and an account of the observations made with it. More; than half the volume however is a reprint of Picard's account of his own operations. S'line observations are also given relating to Aberration. L'.'W. Iii the Paris ,l/////o//r.s for 1740, published in \7\-'2. we have a Memoir entitled / )< lit Mrt'if/tcintc ilc idnx, pruloiK/f'e rcrs If A'//v/, cf <lcx Observations (jut <>nf <'!<' /mfcs JK>UI' <1<<'i'ire I ex fronttercs (.In HoyauiHC, l'<ir M. ('n^sim /)c Thurij, The memoir MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 127 occupies pages 276... 29 2 of the volume. There is an account of it on pages 69... 75 of the historical portion of the volume. The memoir is very important in the history of the subject. Hitherto the accuracy of Picard's base had not been questioned ; but now it was resolved to examine this point. A base not quite coincident with Picard's, but very near to it, was measured five times ; by the aid of a certain length deduced from this it was found that Picard had ascribed to his base a length nearly 6 toises greater than it should have had. In order to leave no doubt on the point, the last measurement was made in the presence of Commissioners from the Academy, at the request of Cassini de Thury him- self. These Commissioners were Clairaut, Camus, and Le Monnier. See La Meridienne de Paris verifie'e, page 36. Bailly implies that Picard's- actual base was remeasured, which as we see was not the case. Moreover, he erroneously states that all the five measurements were made in the presence of the Com- missioners from the Academy. Histoire de I'Astronomie Moderne, Vol. in. page 35. It will be convenient to bring together the various lengths assigned to the degree of the meridian between Paris and Amiens. Picard himself in 1671 adopted 57060 toises; see De la Grandeur et de la Figure de la Terre, page 281. Maupertuis in 1738 by correcting Picard's observations for aberration arrived at 56926 toises. Figure de la Terre, page 126. Maupertuis and his companions in 1740 by new astronomical observations obtained 57183 toises. Degr6 du Me'ridien... First Part, Chapter vill. Cassini de Thury, after the remeasurement of Picard's base, using the amplitude determined by Maupertuis and his com- panions, gave 57074 toises. Paris Me'moires for 1740, page 289. The errors made by Picard in his astronomical and geodetical work had by accident almost balanced each other. The subject is discussed by La Condamine in his Mesure des trois premiers degr&s, pages 239... 258. 237. We return to the memoir by Cassini de Thury. .The memoir is remarkable for being, I presume, the first since the 128 MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. discussion had arisen as to the form of the Earth in which a member of the family of Cassini recognised the oblateness. We learn from page 288 of the memoir that at the north of France the length of a degree of the meridian was found to be 57081 J toises, and at the south of France 57048 toises. Then Cassini de Thury adds : ...ainsi, suivant ces observations, les degres vont en diminuant eii s'approchant de 1'Equateur, ce qui est favorable a 1'hypothese de 1'ap- platissement de la Terre vers les Poles. It may be interesting to compare results given in the present memoir with some given in the earlier work. According to the De la Grandeur et de la Figure de la Terre, page 148, the distance between the" parallels of Paris and Collioure is 360614 toises, the amplitude 6 18' 57", and the mean length of a degree 57097 toises. According to the present memoir, the distance between the parallels of Paris and Perpignan is 350142 toises, the amplitude 6 8' 17", and the mean length of a degree 57045 toises. Again, according to the De la Grandeur et de la Figure de la Terre, page 236, the distance between the parallels of Paris and Dunkirk is 125454 toises, the amplitude 2 12' 9".o, and the mean length of a degree 56960 toises. According to the present memoir, for the same arc the corresponding numbers are ] 25 508 toises, 2 11' 55".5, and 57081.5 toises. 238, It must be observed that the error in Picard's base does not account for the apparent diminution in the length of a degree of the meridian in passing from the equator to the pole which the school of Cassini had deduced and maintained. For in the De la Grandeur et de la Figure de la Terre, which was the main support of this hypothesis, the lengths are all deduced from that of Picard's base; and so the proportions would not be affected by any error in the base. This remark is necessary because the contrary has been asserted, or obviously implied. Thus Bailly says, "l'erreur de cette mesure &oit le nceud de la difficult^:" Histoire de I'Astronomie Moderne, Vol. in. page 38. And on page 169 of the article Figure of the Earth in the Encyclopaedia Metropolitans we read MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 129 " On measuring new bases and making new observations of every kind, the cause of the original difficulty was soon discovered. The measure of Picard's base was erroneous by about Trr^th part of 1UUU the whole, and this error had affected one part only of the arc" The statements which I have here put in Italics do not seem to me supported by the evidence. It is true that in 1739 and 1740 anomalies were revealed which cast suspicion on Picard's measure- ment, and which were explained when that measurement was corrected ; but these were quite distinct from the original difficulty. See La Meridienne de Paris verifiee, page 19. We perceive from this memoir that in 1740 the oblate form of the Earth was fully established and admitted. 239. An edition of Newton's Principia appeared at Geneva in 1739... 1742, edited by Thomas Le Seur and Francis Jacquier. The editors are usually styled Jesuits, and the edition is called the Jesuits' edition. I have already referred to this edition : see Arts. 16, 22, and 232. The commentary on Propositions XVIIL, XIX. and XX. of Newton's third Book does not seem to me very successful ; there are some serious mistakes in it, which occur chiefly in notes marked with an asterisk. It appears from the Monitum and the Editoris monitum, prefixed to the third Book, that these are due to J. L. Calandrinus, to whom Le Seur and Jacquier acknowledge great obligations. I will point out these mistakes. A curious note is given on the words which I have quoted in Art. 26: "Et propterea dico..." The note in effect states that Newton must have had better reason than appears at once obvious for applying the rule of proportion. The note then proceeds to justify the proportion which Newton uses; but the investigation is unsatisfactory for the reason which often applies to approximations, namely, that the calculations are not carried to the same degree of accuracy throughout. Using the letters as in Art. 20 the note asserts that the ratio of the attraction at Q to the attraction of a sphere having C for centre and CQ for radius, is equal to T. M. A. 9 130 MISCELLANEOUS INVESTIGATIONS FROM 17-1 TO 1740. Ll-2. CQ CA ; if the cllipticity e be very small, this reduces to 3 - 2 (1 - e), that is, to 1 + -e : but, as we have stated in Art. 20, 4e the true value is 1 + . o A long note is given on Newton's Proposition XIX., which involves some singular errors ; indeed it seems to me quite extra- ordinary that such a note should have been printed towards the middle of the eighteenth century. The note proposes to investigate the resultant attraction of a homogeneous solid of revolution at the surface ; and it begins correctly by observing that if we take a pyramid with an infinitesimal solid angle, the attraction exerted by a segment of the pyramid on a particle at the vertex varies as the height of the pyramid. Let AB be the axis of the solid of revolution, P any point at its surface, MCN any double ordinate at right angles to AB. The note supposes P to be the vertex of a system of infinitesi- mal pyramids, the axes of the pyramids all passing through the circle generated by the revolution of CM round ( 1 A. The note concludes that the resultant attraction of these pyramids will be in the direction PC: this conclusion is obtained by taking the pyramids in pairs, so that the bases of a pair may be at the op- posite ends of a diameter of the circle; for example, the pyramid which has PM for its axis is combined with that which has PN for its axis. Now it is <|iiite true that such a pair of pyramids will exert a resultant attraction along PC, proridcd t/te two pyra- 'tiwls have C'^mJ infitnte^imal solid (twjlcs : but this important MISCELLANEOUS INVESTIGATIONS FROM 1721 TO 1740. 131 condition is practically forgotten in the note. A laborious calcu- lation is given for determining the resultant attraction of all the pyramids which have their axes passing through the circle formed by the revolution of CM round CA ; but this is of no use, because the bases of all these pyramids will not form a strip of the surface contained between this circle .and an adjacent circle in a parallel plane, though the note implicitly assumes that they will. Again, the language of the note seems to suggest that we are to obtain the attractions exerted on P by all the circular elements like that considered, and add them together. This would however be useless ; for as these attractions are not all in the same direction they would have to be resolved according to fixed directions, and the resolved parts in the same direction added. Again, we are in effect told to obtain the direction of the resultant attraction of the solid in the following manner : Suppose Y a point in AB such that the attractions on Pof the two seg- ments into which the solid is divided by a plane through Y at right angles to AB are equal ; then P Y is the direction of the resultant. This statement is certainly untrue. For instance, if the solid is a sphere, the resultant attraction passes through the centre; but the two halves formed by cutting the sphere by a plane do not in general exert equal attractions on a particle at the surface. It is incidentally stated, .that in the triangle PMN we have (PM + PN) PJV greater than MN*\ but this is not necessarily true. The following extraordinary principle is offered for obtaining the condition of equilibrium of a mass of fluid in the form of a solid of revolution. Let t denote the distance at any point P between the bounding surface and a similar surface indefinitely near, /the attraction at P, y the distance of P from the axis, ds an element of the generating curve at P; then tfyds is to be constant. It is sufficient to observe that in the simplest possible case, that of a sphere, this condition does not hold ; for then t and f are constant, but yds is not constant, except by an arbitrary hypo- thesis. The commentators notice the inaccuracy of Newton, on which I have remarked in Art. 33 ; they assert that gravity at different 92 132 MISCELLANEOUS IXVESTKUTIoNS FROM 1721 TO 1740. places varies inversely as the radius of curvature: " ...gravitates in sin^ulis punetis to rent reciproee ut radii osculatores curva 1 . This is untrue; it would make the gravity greatest at the equator and least at tin 1 poles. Tlie iact is that gravity would vary as the length uftlie normal between the point and the major-axis. The commentators having obtained an expression substantially equivalent to the .. . , which I have Driven in Art. o.">, imme- </> diatelv proceed to take a 3 r 3 for the numerator; but this approxi- mation is not exact to the order which has been retained. 1 should add, however, that in their next note there is a correct analytical investigation of the matter. O 240. We mav next advert to a memoir entitled Deter minatio xc(ctiur (ii'ddmun PtiruUeloruin ^"Kquutoris et Meridiinii.f.Auctore C. JV. de Winsheiin. This is contained in tlic Cunimentarii Academic?.. .retr<>i)<>Iit(tn<i>, Vol. xn. which is for 1740; the date of publication is 17-"<0. The memoir occupies pa^cs 222. ..240 of the vulume. Hen- we have Tables ^ivin.n 1 the K-n^ths of a decree of the meridian and of a derive of longitude !;i various latitudes, for a sphere, and for an oblaium in Avliirh ih( > ellipticity is - %> - a This ellipticity is i'oiind from the La})land degree of .")74.*>S toises, and Picard's taken at .">71 ^o toises : see Alt. 2o(J. Winsheim ascribes to Euler the rule which he uses for calculating the Tables with respect to the oblatum. CHAPTER IX. MACLAURIN. 241. MACLAURIN'S researches on Attractions first appeared in his Essay on the subject of the Tides, which gained a prize from the French Academy in 1740 ; see Art. 232. These researches are reproduced in an enlarged and improved form, in Maclaurin's work entitled A Treatise of Fluxions, Edinburgh, 1742. The work is in two quarto volumes ; it contains Title Pages, a Dedication to His Grace the Duke of Argyle and Greenwich on two pages, a Preface on six pages ; then the text on 763 pages, and a page of Errata : there are XL Plates. The Treatise of Fluxions embodies much of the analysis and mechanics of the period. Maclaurin touches on the equilibrium* of fluids in his pages 409, 410. We may infer that he had a cor- rect idea of what we now call the differential equation to the surface of a homogeneous fluid in equilibrium under given forces. 242. The part of the Treatise of Fluxions with which we are concerned, occupies pages 522... 566, which are in the second volume. Maclaurin shews that the attraction of a homogeneous cone with a given infinitesimal solid angle on a particle at the vertex varies as the length of the cone ; and that the same result holds for a frustum of the cone ; the particle being still supposed at the vertex of the cone. See his Article 628. Then his Article G29 draws an important inference, which Newton had given in the first corollary to his Proposition 87. Maclaurin says : The forces with which particles similarly situated with respect to similar homogeneous solids gravitate towards these solids are as their Distances from any points similarly situated in the solids, or as any of 134 MACLAURIN. their homologous sides. For such solids may be conceived to be re- solved into similar cones, or frustums of cones, that have always their vertex in the particles, and the gravitation towards these cones, or frustums, will be always in the same ratio. In future, if nothing is said about the density of the attracting body it is to be understood to be a homogeneous body. 243. Maclaurin shews in his Article 630, that a particle will be in equilibrium if it is placed at any point within the hollow part of a shell, the surfaces of which are concentric, similar, and similarly situated ellipsoids of revolution; the demonstration is the same as Newton's : see Art. 13. 244. Let the attraction of an ellipsoid of revolution on any constituent particle be resolved into two components, one perpen- dicular to the axis, and the other parallel to the axis ; then the former component varies as the distance of the particle from the axis, and the latter component varies as the distance from the plane of the equator. Maclaurin demonstrates these theorems, first formally stated by himself, by a beautiful geometrical pro- cess: see his Articles 631... 634. Clairaut preserves the essence of Maclaurin's demonstration: he says, "Cette methode m'a paru si belle et si savante...": see Figure de la Terre, pages 157... 170. Suppose that X denotes the constant coefficient for the com- ponent attraction parallel to the axis, and /j, the constant coefficient for the component perpendicular to the axis ; then, by some general reasoning, Maclaurin arrives at the result that the product of X into the square of the polar axis is less or greater than the product of p, into the square of the equatorial axis according as the ellipsoid of revolution is oblate or oblong : see his Article 635. 245. Let there be an ellipsoid of revolution; let 2a be the equatorial diameter, and 26 the polar diameter. Suppose the ellipsoid to be fluid ; and besides the mutual attractions let there be at every point any other force perpendicular to the axis varying as the distance from the axis, and any other force parallel to the axis varying as the distance from the plane of the equator: MACLAURIN. 135 the necessary and sufficient condition for equilibrium is that a must be to b, as the resultant force at the pole is to the resultant force at the equator. This theorem can be demonstrated imme- diately by the aid of the well-known equations for the equilibrium of a fluid. Maclaurin, however, was not in possession of these equations ; so that he adopted a different method. He says in his Article 636 : To demonstrate this proposition fully, we shall shew, 1. That the force which results from the attraction of the spheroid and those extra- neous powers compounded together acts always in a right line perpen- dicular to the surface of the spheroid. 2. That the columns of the fluid sustain or ballance each other at the center of the spheroid. And 3. That any particle in the spheroid is impelled equally in all di- rections. He gives his demonstrations in his Articles 637, 638, 639. Maclaurin then was in this position : there was as yet no theory of fluid equilibrium which indicated what conditions were sufficient, so he shews that all the conditions which had then been recog- nised as necessary for equilibrium would be satisfied in the case supposed. He easily demonstrates the first condition, which, as we know, was given by Huygens: see Art. 53. Maclaurin's second condition is a particular case of his third, and was given by Newton : see Art. 23. The meaning which Maclaurin attaches to his third condition is the following : Take any definite point within the mass ; draw from this point a straight line to the sur- face in any direction ; let this straight line be the axis of a column of given infinitesimal section : then the attraction on the column resolved along the column, is independent of the direction. Maclaurin, however, only demonstrates this for the case in which the direction is in the meridian plane of the definite point ; he says that " in like manner, it is shewn " that the result is true for columns not in the meridian plane: but it is not obvious how he would have proceeded. The result can be obtained very easily by modern methods. Maclaurin's third condition is thus an extension of Newton's principle of balancing columns, any point being taken instead of the centre, at which the balancing is to hold. Huygens had briefly alluded to this extension : see Art. 55. 136 MACLAURIN. 246. This extension of Newton's principle of balancing columns seems to have been considered important at the time. D'Alembert says on page 14 of his Essai...de la Resistance des Fluides : Quoique le Principe de 1'equilibre des Canaux rectilignes, soit comme Von voit, une consequence tres-nat/urelle de la pression des Fluides en tout sens ; cependarit je dois reconnoitre ici, que feu M. Maclaurin est le premier qui ait fait usage de ce Principe, et qui 1'ait applique a la recherche iinportante de la Figure de la Terre. Voyez son Traite des Fluxions, art. 639, et son Traite de Causa Fluxus et Refluxus maris, Paris, 1740. See also D'Alembert's Traite'... des Fluides, second edition, page 49. 247. In Maclaurin's Article 637, we have the important result which we have noticed in our account of Stirling; namely, that when rotating fluid in the form of an oblatum is in relative equilibrium the gravity at any point of the surface varies exactly as the length of the normal between the point and the plane of the equator ; see Art. 153. This result had however been com- municated to the Royal Society by Simpson, in 1741, before the publication of Maclaurin's Fluxions: see the preface to Simpson's Mathematical Dissertations. Simpson seems to claim priority for himself; but he overlooks the fact that Maclaurin had previously given the result in his prize essay on the Tides : it is the Theorema Fundamentale of the essay. It follows immediately from conic sections that instead of the gravity varying as the length of the normal between the point and the plane of the equator, we may take the length of the normal between the point and the axis of revolution. 248. Maclaurin, in his Article 640, states the conclusions which he had thus demonstrated respecting the problem of Art. 245. Among them we may observe that he says, surfaces similar, simi- larly situated, and concentric with the bounding surface " will be level surfaces at all depths." This is the first mention I find of level surfaces ; the essential property of a level surface is that the resultant force at any point of the surface is in the direction of the normal to the surface at that point. MACLAURIN. 137 B'Alembert in his Essai. ..dela Resistance des Fluides, page 202, says: ... M. Maclaurin, le premier qui ait parle de ces couches ...... aux- quelles la pesanteur est perpendiculaire, et qu'il appelle surfaces de niveau. . . . 249. Maclaurin now applies the results obtained for the general problem of Art. 245 to the particular case of the relative equilibrium of a revolving fluid. He says in his Article 641 : It appears therefore that if the earth, or any other planet, was fluid and of an uniform density, the figure which it would assume in consequence of its diurnal rotation, would be accurately that of an ohlate spheroid generated by an ellipsis revolving about its second axis, as Sir ISAAC NEWTON supposed. Here, Maclaurin says more than he was justified in saying; he had not proved that the planet would assume the form of an oblatum, but only that this form is a form of relative equilibrium. See Art. 168. The proposition really investigated was first established exactly by Maclaurin ; as we have stated, Stirling and Clairaut had given approximate investigations of it : see Arts. 156 and 163. 250. Maclaurin now proposes to calculate the attraction of an ellipsoid of revolution at the pole or at the equator. He begins with a lemma which forms his Article 642. Let a slice of an attracting body be formed by two planes, both containing the attracted particle, and inclined to each other at an infinitesimal angle : then the lemma shews how to calculate the attraction of the slice resolved along a given direction in one of the planes. 251. Before discussing the attraction of an ellipsoid of revo- lution, Maclaurin considers that of a sphere in his Article 643. The following general result is obtained : Let C be the centre of a circle, P any external point in the plane of the circle. From P draw any straight line cutting the circumference of the circle at L and If; and let a solid be formed by the revolution round PG of the smaller segment of the circle cut off by LM. Then the attraction of this solid on a particle at P varies as 7-y . (1 U) 138 MACLAURIN. This may be easily verified by the aid of the general expression given in Art. 4. The formula is very remarkable ; it does not involve the radius of the sphere ; that is, if LM is constant, we get the attraction constant whatever may be the value of the radius. The result was generalised by Legendre, as we shall see, in his third memoir. 252. Maclaurin then in his Articles 644... 647 investigates accurate expressions for the attraction of any ellipsoid of revolution on a particle at the pole or at the equator. The investigations are conducted in the manner of the time by representing the attractions by the areas of certain curves, and finding the areas by the method of fluents. The results agree with those obtained by analysis, and presented in modern works on Statics. Maclaurin's processes are remarkable specimens of ingenuity, considering the date of their publication; but they will not be very interesting to a modern reader. 253. Maclaurin says in his Article 647 : ... What lias been shown concerning the gravity at the pole... agrees with what was advanced long ago by Sir ISAAC NEWTON and Mr. COTES, who contented themselves with an approximation in determining the gravity at the equator, which is exact enough when the spheroid differs very little from a .sphere. The approximations proposed lately for this purpose, Phil. Trans. N. 438 and 445. are more accurate ; and Mr. STIRLING after determining the gravity at the equator by a converging series, since found that the sum of the series could be assigned from the quadrature of the circle. I do not know what is intended by the reference to Mr Cotes, Of course Cotes, as editor of the Principia, may be supposed to have accepted some of the responsibility which would otherwise have fallen on Newton alone: but Maclaurin's words seem to imply that Cotes had made some investigations of his own. The paper in the Philosophical Transactions, Number 438, is that by Stirling, of which we gave an account in Chapter V. ; and the paper in the Philosophical Transactions, Number 445, is that by Clairaut, of which we gave an account in Arts. 103... 106. I do not know wluit Mackurin means by the words "and Mr Stirling.. .circle." This passage from Maclaurin was quoted, mul the difficulty MACLAURIN. 139 as to its meaning noticed, by the late Sir J. W. Lubbock : see page 24 of his work cited in Art. 233. I do not know whether the conjecture may be considered plausible that Maclaurin wrote Stirling by mistake for Simpson. It appears from the preface to Simpson's Mathematical Disserta- tions that his researches on the Figure of the Earth were read to the Royal Society in March or April, 1741 ; and what Maclaurin says with respect to Mr Stirling is not unsuitable to the investi- gation we find in Simpson's work, except that Simpson does not restrict himself to a point at the equator, but takes any point on the surface. 254. Maclaurin proceeds in his Articles 648... 652 to one of the most important of his investigations, remarkable as forming a large part of the theorem which now usually bears the name of Ivory, though it was substantially first demonstrated by Laplace. Maclaurin's theorem is as follows in modern language : Let there be two confocal ellipses, and let them both revolve round their major-axes, or round their minor-axes, so as to gene- rate two ellipsoids of revolution : then the attractions of the two ellipsoids on the same particle external to both will be as the volumes, provided the particle be on the prolongation of the axis of revolution, or in the plane of the equator. Two such ellipsoids may be called confocal ellipsoids of revolution. Legendre shewed that the theorem was true for any position of the external particle. The general theorem demonstrated by Laplace is as follows : If there be two confocal ellipsoids, that is, ellipsoids which have the same foci for their principal sections, their attractions on any particle external to both will be as their volumes, that is, will be the same in direction, and in amount will be as their volumes. The simplest statement in modern language is this : The potentials of confocal ellipsoids on a given external particle are as their volumes. Maclaurin in a later Article, namely 653, gave so much of this general theorem as consists with the limitation that the particle must be on the prolongation of an axis of the ellipsoids. Ivory merely supplied an improved form of demonstration to Laplace's theorem ; and combined it with the fact that inside an ellipsoid, along any radius-vector, the attraction varies as the distance from the centre. 140 MACLAURltf. Maclaurin's Articles 648 and 649 contain his demonstration for the case in which the external particle is on the prolongation of the axis of revolution. These Articles may be read without diffi- culty, apart from Maclaurin's other investigations, by those who are desirous of seeing a specimen of his own processes. 255. It is easy to translate into modern language the essence of Maclaurin's demonstration, Let 2a and 26 be the axes of an ellipse ; let the ellipse revolve about the axis of length 2a, and thus generate an ellipsoid of revo- lution : required the attraction of the ellipsoid on a particle which is on the prolongation of the axis of revolution at a distance c from the centre. Let r be the distance of the attracted particle from an y point of the ellipsoid ; let 6 be the angle between r and the axis of revolution. We see in the usual way that the attraction is found by integrating with respect to r and 6 the expression %7rrdr r sin 9 cos dd ~r*~ Integrate with respect to r and we obtain 2?r (r t - rj sin 6 cos 6 dd, where r 2 and r t are respectively the greatest and the least values of the radius-vector drawn from the attracted particle to the ellip- soid at the inclination 6 to the axis of revolution. Hence r 2 and r t are the roots of the quadratic equation (r cos<9-c) 2 r 2 sin 2 _ 1 ~~cT ~W~ and thus we shall find that _ 2ab V(6 2 cos 8 9 + a 2 sin 2 6 - c' sin 2 0) Tr cix?~0~+a' sin 2 _ 2a V{6* + (a* - fr* - c 2 ) sin 2 6} Now let there be a second ellipsoid of revolution, having the foci of its generating ellipse in the same position as before ; and MACLAURIN. 141 let accented letters be used to denote the analogous quantities ; so that ,_ , _ 2a'V V(5' 2 + (a' 2 - V* - c 2 ) sin 2 0'} r * r i = &' 8 + (a' a -&' 2 )sin 2 0' Since the foci of the generating ellipses are coincident, we have a? 6 2 = a' 2 b' z , whether the ellipsoids are oblate or oblong. Assume sin & = -j- sin 9 ; then we see that b al ,d therefore fc Thus the attractions of the corresponding elements of the two ellipsoids resolved along the direction of the axis of revolution are in the same proportion as the volumes of the ellipsoids ; and so the resultant attractions of the whole ellipsoids will be in that proportion. It will be observed that on our assumption r a ' r{ and r 2 - r t vanish together ; so that our elements always correspond. If the density of one ellipsoid is not the same as the density of the other, then the attractions will of course be in the ratio of the masses instead of the ratio of the volumes. This remark will be obviously applicable in some subsequent Articles. Maclaurin's own investigation in his Art. 648 applies to his figure 292, which is drawn for an oblatum ; but the figure may be drawn for an oblongum, and it will be found that the investiga- tion is equally applicable. In Maclaurin's investigation the point Pis o?i the larger ellipsoid ; but still this involves the result in as general a form as we have stated it. 256. Maclaurin's Article 650 consists of three sentences; it would have been advantageous, for the sake of clearness, if they had been printed as three distinct paragraphs : the last sentence most certainly should have been separated from the others. In the first sentence Maclaurin gives an expression for the attraction of an oblatum on an external particle which is situated 142 on the axis of revolution: this follows from his former results, which we have noticed in Arts. '2~t'2 and '2 ">.">. In the second sentence Maclaurin gives the corresponding ex- pre>sion for the attraetion of an obluiigum. The third sentence is very remarkable. It has heen shewn that tlie attraction of a homogeneous ellipsoid of revolution on an external particle which is situated on the axis of revolution, varies as the mass, so long as the generating ellipse keeps its foci fixed : now suppose an ellipsoid of revolution, not homogeneous, hut made up of shells, each shell bring bounded by cenfocal ellipsoids <if revolution, and the density bring uniform throughout each shell, but varying in any manner from shell to shell : then tin- attraction of this heterogeneous ellipsoid on an external particle .-ituated on the axis of revolution is to the attraction of a homogeneous ellipsoid of the same size as the mass of the former is to the mass of the latter. This is the first appearance of these coi/fucal shells, which play an important part in modern works on Attraction. 2-">7. Maclaurin now proceeds in his Articles (jol, (J-~)2 to the case in which the attracted external particle is in the plane of the equator of the attracting ellipsoid of revolution. He uses a most ingenious artifice by which this case is made to depend on that already considered, in which the attracted particle is on the prolongation of the axis of revolution. We will translate his process into modern language. ',i'~ ?/" -4" z~ Let the equation to one ellipsoid of revolution be + ' =1, a c" and the conation to another ' ., +' -=1. Suppose the gene- a" ('" rating rllipsrs to havr the same foci ; thru, wlirtlirr the ellipsoids arc ublatu or oblong, a 2 <' = n'~ <'". Suppose llir second ellipsoid t<> he the larger. We propose to compare tin- attractions of th'-M- ellipsoids on a particle which is on the eiju;itor of tin- larger ellipsoid; the co-ordinates of the particle mav be i.ak'ii to br 0, 0, c . \Ve shall .shew that the MACLAURIN. 143 Let C denote the centre of the ellipsoids, and P the position of the attracted particle. Let two planes pass through CP, and make with the axis of y the angles 6 and + 0, respectively : we will call these planes the first pair of planes. Let two other planes pass through CP, and make with the axis of y the angles & and & + 6' respectively : we will call these planes the second pair of planes. The volume comprised between the first pair of planes and the first ellipsoid we will call the element of the first ellipsoid ; the volume comprised between the second pair of planes and the second ellipsoid we will call the element of the second ellipsoid : each element then consists of two wedge-shaped slices. We shall shew that when a suitable relation is made to hold between 6 and &, the attractions of these elements on the particle at P are as their volumes. The relation between 6 and & is found by assuming that the ellipses which form the boundaries of the elements shall be confocal Thus we have r* - c 2 = r' 2 - c' 2 , aV ' 2 ' 2 where r 2 = ., , .. . . a/1 , and r a* cos 2 6 + c 2 sin 2 ' a' 2 cos 2 ff + c" 2 sin 2 0' * Since a 2 c 2 = a' 2 c' 2 , we obtain c 2 sin 2 _ c' 2 sin 2 0' a 2 cos* + c 2 sin" 6 ~~ a 2 cos 2 ^ + c' 2 sin 2 0' : this is the relation between 6 and 6'. It is obvious that to the limits and = for 9 correspond the same limits for 6'. Suppose now that a solid were formed by the revolution round CP of an ellipse having C for centre, 2c for the axis of revolution, and 2r for the other axis. Let F denote the attraction of this solid on the particle at P. Then it is obvious that ultimately the attraction of the element of the first ellipsoid on the particle is 7T Also suppose that a solid were formed by the revolution round CP of an ellipse having C for centre, 2c' for the axis of revo- lution, and 2r' for the other axis. Let F' denote the attraction of this solid on the particle at P. Then it is obvious that ultimately 144 MACLAUniN". the attraction of tin- <-lemc'nt <>f tl/<' second ellipsoid on the particle . M' r . is F 7T Therefore if/'andy denote tlie attractions of the elements, we / _ F. 80 f ~?"8tf" Now, as we have seen in Art. 2 ">.">, Maclanrin liad shewn that F re F ~ i-'V ; / rcW therefore ' ., = ,., >*/ J r'coo But rcO represents the area intercepted by tlie first pair of planes from the ellipse ., -f* -4 = 1 : and r'W n-pri-sents the area inter- a~ c~ eeptt/d hy the scrond pair of plain s from tin- ellipse ,, -f ' ,, = 1. Thus \ve see that f is to/" as the volume of the element of the first ellipsoid is to the volume of tlie element of the second ellip- snid. .And as tins proportion holds for every corresponding pair of elements it holds lor tin.- entire ellipsoids; which is what we had to demonstrate. 2">.s. The process may he easily extended to the case in which the- ellipsoids are not of revolution, as Maclaurin himself indicates in his Article (j.">3. Let the e(|iiations to the 1 ellipsoids be X II Z~ I ,/ I I , ' a -T i + = ! > '2 + /'* + '- ' ' //- Ir c~ a l>" c and lei the principal sections of the ellipsoids be confocal, M> that Th.- relation between and 0' will then he found from the cond it ion /"' c' /''" < ", As before, w<- shall find that to tin 1 limits and ) tor 9 corre- MACLAURIN. 145 spond the same limits for '. Then the investigation and the result will be as in the preceding Article. 259. Thus in the attraction of homogeneous ellipsoids Maclaurin's position was as follows : he solved completely the problem of the attraction of an ellipsoid of revolution on any internal particle; and with respect to an external particle, he obtained for ellipsoids, not necessarily of revolution, the theorem of Laplace, so far as relates to a particle on the prolongation of an axis of the ellipsoids. All this w'as exactly demonstrated. Maclaurin states also something more as approximately true in his Article 654. The statement amounts to this, that the theorem of Art. 254 is true " either accurately or nearly when the spheroids differ little from spheres/' when the attracted particle has any position. He gives no detail as to the investigation of this result; but merely says it may be deduced from his Article 653. We know now that the theorem is exact and not merely an approx- imation ; and, as we have stated, the demonstration was first given by Legendre, and the theorem is a part of Laplace's general theorem. 260. The extent to which Maclaurin carried his investigations was under-estimated by many of the succeeding writers. He was supposed to have merely enunciated the result which we have noticed in Art. 258, whereas he really demonstrates it : he says "it will appear in the same manner..." and it is clear from an examination of his context that this is the case. The erroneous account will be found in the following places: D'Alembert, Opuscules Mathtfmatiques, Vol. vi. 1773, page 243 ; Lagrange, Berlin Me'moires for 1775, page 279; Laplace, Th^orie...de la Figure elliptique des Planetes, 1784, page 96 ; Legendre, Me'moires ...par divers Savans, Vol. x. 1785, page 412. Laplace, Me'canique Celeste, Vol. v. page 9. Plana in Crelle's Journal fur... Maihe- matik, Vol. xx. page 190. According to the catalogues of book- sellers, it appears that Maclaurin's Fluxions was translated into French, so that there is less excuse for the error. I suppose that D'Alembert went astray, and the others followed in succession without examination. Chasles is correct ; he says that Maclaurin T. M. A. 10 146 MACLAURIN. did demonstrate his theorem, and he points out the error in this matter made by D'Alembert, Lagrange, Legendre, and others : see the Memoires...par divers Savants, Vol. ix. 1846, page 632. The error is also noticed by Dr F. Grube in a paper in the Zeitschrift fur Mathematik und Physik, Vol. XIV. Leipsic, 1869, page 272. On the other hand, some recent English writers have gone to the opposite e'xtreme, and given to Maclaurin more than his due, by ascribing to him in effect the entire theorem called Ivory's, but more strictly Laplace's ; see Natural Philosophy, by Thomson and Tait, Vol. I. page 392, and Routh's Rigid Dynamics, 2nd edition, page 421. 261. It will be convenient to give the results obtained by Maclaurin as to the attraction of an oblatum on an external par- ticle which is in the plane of the equator, or on the prolongation of the axis of revolution. Let P be the position of an external particle which is in the plane of the equator. Let F be the focus of the section of the oblatum made by the plane which contains P and the axis of revolution. Let C be the centre, CA and CB the semi-axes of the section. With F as centre, and a radius equal to CP, describe a circle cutting CB produced at D. With D as centre, and DF as radius, describe the arc FO, and with D as centre and DC as radius, describe the arc CS. MACLAURIN. 147 Then Maclaurin obtains for the attraction on a particle at P the expression 2.CB.CA* are&FCO ,,/, -x- rp - U-C \J L And for the attraction on a particle at the point D on the prolongation of the axis of revolution, he obtains the expression *' C A \(CF-CB). If we multiply these expressions by 2?rp, where p denotes the density, they will be found to agree with those given in modern works on Statics when we suppose P to be on the surface ; and the case where P is not on the surface may be deduced from that where P is on the surface, by Maclaurin's theorem of Art. 254. The presence or absence of such a factor as 2?r merely depends on the choice we have made of the unit of attraction. Put a for CA, and ae for CF\ also put r for CP in the first expression, and r for CD in the second; then, introducing the factor 2irp, our expressions become : 2-TTp A/(l ~ 2 ) ( -i ea ae V(** 2 ~ 2 ft 2 )) t-i \ a \T sm - -> (1), e r r } and ' Jea-rtair 1 '-^ (2), e* \ r) so that (1) applies to the particle in the plane of the equator, and (2) to the particle on the prolongation of the axis of revolution. It will be useful for us to collect here some, obvious deductions from (1) and (2). The attraction at the equator is obtained by putting a for r in (1) ; and the attraction at the pole is obtained by putting a V(l e 2 ) for r in (2). Let x and y be the co-ordinates of any point on the surface of the oblatum, measured from the origin C parallel to CA and CB respectively. Then, by Art. 244, combined with the values of the attraction at the equator and at the pole, to which we have just alluded, we obtain for the attractions at the point (#, y), resolved parallel to CA and CB respectively, 102 14S MACLAURIN. and e~ If we expand these and neglect e 4 and higher powers of e we obtain respectively By expanding their second factors in powers of e, the expres sions (1) and (2) become respectively and In the expressions (1) and (2) change a into <r -f S<7, and sub- tract the original values; thus we obtain the attraction of a shell bounded by similar, similarly situated, and concentric oblata, on an external particle in the plane of the equator < r on the prolon- gation of the axis: supposing Sa so small that all powers beyond the first may be neglected, the results are respectively and e r + a Maclanrin, subsequently, in his Article's ()()8 and (!(!.0, gives without demonstration, in a geometrical form, results which are O ? to these. 202. Maolaurin, in his Article (;.">.">, ;ip]lies his results to find the condition for the relative equilibrium of an oblatum of fluid rotating round the minor axis. Let a be the semi-axis major, and c the excentricity. Let A" denote the attraction at the equator, and Y the attraction at the pole. Then we obtain X by putting t( for r in trie expression (1; of Art.2(il, and we obtain 1" by putting '/ \ 7 ( 1 J) tort 1 in the expresMon (2). Thus we find MACLAURIN. 149 X Suppose that jX denotes the value of the centrifugal force at the equator; then for relative equilibrium we must have, by Art. 245, __ ,r^ ^fT^S^S y* _ therefore Y Put for Y its value, and this* becomes . 3 {sin" 1 e - e V(l - e 2 )} - 2e 2 sin" 1 e sin 1 e e V(l O These expressions are exact. By approximation we obtain r- 1 - I >j o -j- f - v> o oo ^ = rT^ ~ s 1 h !0 e *~56 e f 48 Maclaurin gives these approximations as far as e 4 inclusive. By reversion of series we obtain so that when the oblatum differs very little from a sphere we may take 5; 2 Maclaurin then says, "in this case the excess of the semi- diameter of the equator above the semiaxis is to the mean semi- diameter nearly as" 5j is to 4 - ^ . By the mean semi-diameter 150 MACLAUKIN. he intends half the sum of the polar and equatorial radii. Taking 1 for the equatorial radius, we have V(l } for the polar radius; then the ratio of the difference to the half-sum is expressed exactly by - j 1 "ffijff* . If we wish to be correct only to the first power of e 2 this becomes -= . If we wish to be correct to the second power of <? this becomes . We might use other forms which would coincide \ with this as far as the second power of e*. For instance, we have the ratio exactly equal to T -^ ^ , and thus to the order I ~T~ V \-*~ ~~ ^ )i of e 4 we get 2 ; and then we may put this to the same f' I P \ order in the form -^ ( 1 + -^ j . M 2e 2 2 Taking the form -j 9 and putting for e 2 , 1 + y 4-^ * ^ + 7 with Maclaurin 263. Maclaurin shews how the value of e 2 for the Earth, supposed homogeneous, may be deduced from the measured length of a degree of the meridian in any latitude, and the measured length of the pendulum which vibrates in a given time in that latitude: see his Articles 656... 658. He shews in his Article 657 that the radius of curvature in the ellipse varies as the cube of the length of the normal terminated by the major axis ; he was, probably, the first to demonstrate this : see the Mfaa- nique Celeste, Vol. v. page 6. Maclaurin also shews how the value of e* may be deduced from the distance and the periodic time of a satellite revolving in the plane of the equator : see his Articles 659 and 660. MATLAURIN. 151 Maclaurin in his Articles G61...665 obtains numerical results with respect to the Earth, supposed homogeneous. He does not determine strictly the value of the quantity we denote by j ; but he finds as the value of the ratio of the centrifual force at the equator to the force of gravity at Paris, and ^ as the value of the ratio of the centrifugal force at the equator to the force of gravity at the Polar circle. For the ratio of the axes of the Earth he obtains a result practically equivalent to Newton's value of 23Q to 229. Maclaurin shews, howeyer, that this result is not consistent with tfyat obtained by means of the observations of pendulums in various latitudes; nor with that obtained from the measured lengths of a degree of the meridian in France and in Lapland : both these methods gave for tfye ellipticity a larger value than - . ^oO We have now more accurate observations and measurements than those accessible to Maclaurin ; and we know that the true value of the ellipticity is about -^--. . oUO 264. Maclaurin then proposes to treat the Earth as not uni- form in density. In his Article 666 he supposes that there is more matter at the centre than is consistent with the hypothesis of uniform density ; and in his Article 667 he supposes that there is less matter at the centre." He concludes that both these sup- positions are inadmissible, as not agreeing with facts ; for, relying on the French and Lapland arcs, he considered that the ellipticity must be greater than . In his investigations he does not shew that there will be relative equilibrium in the supposed fluid mass; but he shews that if there be relative equilibrium, certain relations will exist between the lengths of the polar and the equatorial diameters. Maclaurin's investigations do not appear quite satisfactory; let us take his Article 667. With the notation of Art. 262 we L32 MACLAU1UX. have XjX for the gravity at the equator, and Y for the gravity at the pole. The ratio of the difference to the half-sum is 9 Y-X+jX ' Y+X-jX* Now for relative equilibrium we must have rv(i-') = ^(i-j); substitute, and we find that the above ratio becomes As we have seen in Art. 262, this result can be put in various approximate forms. Now Maclaurin supposes that matter is removed from the centre of the oblatum, so as to diminish the attraction at the equator by a certain fraction of the mean attraction ; we shall denote this fraction by \, and the mean attraction by G. The \ ri attraction at the pole will be diminished by -- -- ^ . The ratio of J. o the centrifugal force to the attraction at the equator is supposed to remained unchanged. Thus the gravity at the equator is (X \G) (1 /), and at the pole is Y-- - - Z \G. The ratio of the difference to the half-sum -L ~~ L/ IS Y+X(l-j)-\G\^,+ l-j\ Maclaurin considers that this is approximately equal to ^ . ";,. ; 4 -r A. + *-J A, and this is less than ' which he takes for the approximate value of the ratio before the matter was removed from the centre. But these statements are liable to the objection which is fatal to so nianv approximate calculations; the investigation is not true to the order of the small quantities which are retained. Put \ y (Y+X] f<>r G; and observe that Y^(\ -V)=A"(1 -j). Then the ratio after the matter is n-nmvcd fnun the centre is accurately MACLAURIN. 153 \( i i i ( i ~ + " 1 - If we neglect powers of e z and j above the first the numerator of this fraction becomes e a 2X (e 2 + j) ; and the denominator be- comes 2 + -^ ^ ( 2 + ^ + ^ j (2 + e a y), that is, to our order of approximation 2 + ^ X(2 + -~-J. If we now put -^ for e z , we obtain for the ratio . v . Thus we see that Maclaurin is wrong in his denominator. There is, however, a very serious objection to the process just given. If Maclaurin retained the term in j\ in the denominator, he ought to have carried on the approximations in the numerator to a higher order ; for instance, e* ought to have been retained : and then when the value of e z in terms of j is substituted in the numerator the square of j must be retained. But, in order to determine in a satisfactory manner how far the. approximations are to be carried, we must make some hypothesis as to the value of X. Suppose, for instance, that X is ^ or -r ; then j\ will be of the same order as j ; and in the numerator of the ratio we shall have to retain the squares of e* and j t and the product e*j.' But if we suppose that X is of the same order as^", and retain the term j\ in the denominator, then we must make our numerator accurate to the third order of small quantities, and our denominator accurate to the second order, considering e z or j as of the first order. I have taken G = = ( Y+ X) as Maclaurin's words certainly imply. I do not retain his notation nor his language; but use what I find most convenient. Maclaurin himself, in his Art. 666, explains that in what follows he uses gravitation for the excess of gravity above centrifugal force : so that his gravity corresponds to my attraction, and his gravitation to my gravity. 154 MACLAURIN. It is possible however that, with Maclaurin, G=-^(Y+XjX). This meaning of G makes YJ(lj)=2G ^ . //^ ^ ' and the ratio of the difference of the polar gravity and the equa- torial gravity to the half-sum becomes accurately l_V(l-6-) v f 1 Instead of the expression 4 + -^ 4\ -~- j\, which we obtained before for the denominator by approximation we should now have 4 4X 3/X, which is still different from Maclaurin's result. However, though Maclaurin's process is very unsatisfactoiy, his conclusion is true that the ratio of the difference to the half- sum of the gravities is diminished by removing matter from the centre. The best way of shewing this, is to start from the alger braical fact that - *T- is less than *- if -, is greater than - . Ac- q-q q q q T=*-( I -J>. cordingly we have only to shew that *. is greater than ^ +1 " y . ^r'j this reduces to shewing that 1 j is less than J. + v(l e ) t jjz ^r , wijich is obviously true. There would, however, be little interest in ascertaining that the ratio is diminished without any estimate of the amount of diminution ; but, in order to form such an estimate, it would be necessary to make an hypothesis as to the value of X, and then to approximate to a suitable degree of accuracy. Hitherto in this Article we have not paid any regard to the supposition that the oblatum is fluid; but let us now adopt that supposition. Maclaurin finds by Newton's method of balancing columns that when matter is removed from the centre, the polar MACLAURIN. 155 diameter will be diminished, and the equatorial diameter increased, and so the excentricity increased. The process is not satisfactory ; for Maclaurin does not shew that the fluid can remain in equili- brium when matter is removed from the centre : and in fact we now know that it will be necessary to make some fresh hypothesis. We may suppose that there is a solid spherical nucleus, surrounded by a fluid of greater density. In this case it will be found that relative equilibrium will subsist, when the bounding surface is an oblatum of certain excentricity ; and this excentricity is greater than when the body is entirely fluid and homogeneous. But the 2 value of X cannot be taken quite arbitrarily: it must fall below . 5 The problem in fact was solved by Clairaut in the more general form of a central nucleus which is not a sphere but an ellipsoid of revolution, having for its axis of revolution the axis of rotation, See his Figure de la Terre, page 219. We will briefly solve the problem, when the nucleus is spheri- cal, in the modern way. Let M denote the mass of the body, supposed entirely fluid and homogeneous ; then \M is the mass which is supposed to be removed, so as to make the central nucleus less dense than the fluid. We may consider that the attraction at any point of the fluid is produced by the action of the whole ob- latum of fluid, diminished by the action of the sphere of mass \M, Take the axis of z for that of revolution. Let o> be the angular velocity. The attraction of the oblatum at the point (x, y, z] parallel to the axes will be Ax, Ay, Cz, respectively, where A and C are constants. The attraction of the sphere will be 3- , % , and j- respectively, where r 2 = v? + y* + z*. Hence the equation to the surface of the fluid must be Suppose 2a and 2c, the equatorial and polar diameters ; then Ac? G>V \M we get ~ --- o- + - - = constant, 156 MACLAURIX. and ., H --- = constant ; c therefore by subtraction Aa* - (V - o>V + 2XJ/ f - - -) - 0. V" cj Now by hypothesis we have - Cc* - Ac? - + fi\M l - l ] = 0. so that / If we suppose that e is very small, we find by Article 262 that approximately 11 aml - so that e 2 _ A - > 1 ;>X ~ - x 1 - -9" It is obvious that if we suppose e 2 and J to be of the same order of magnitude, this process is not satisfactory for every value 2 of X : for instance, X must not be nearly equal to ^ . And if X is itself of the same order as e 2 and f /, the result is not admissible, for then we ought to have retained e 4 and e*j as well as j\ and e 2 X. We may accept the investigation as sufticiently accurate for such cases as X , or X=- ; and we see that the excentri- .> 10 city is greater than for the case of the oblatum entirely iluid and homogeneous: so far then we agree with Maelaurin. Maclaiirin, liowev-r, acrts, that in consequence of this in- cre.'ise of \}\c -\cent ricit v, the ratio of the dilfei'ence of the gravi- ties to their half-.-um is ]-eii<lrred still les.> than it was before we MACLAURIN. 157 adopted the supposition of fluidity. This is a mere assertion unsupported by evidence. So far as the influence of the removal of central matter is concerned, we may admit that the increase of the excentricity tends to bring the polar gravity and the equa- torial nearer to equality ; but, on the other hand, considering all the other matter as forming a homogeneous oblatum, we see that the increase of the excentricity tends to bring the polar gravity and the equatorial further from equality. Thus, to obtain the actual result, we must strike a balance between opposing influ- ences ; and this Maclaurin has not done. We can easily submit the question to calculation. Before the 2 hypothesis of fluidity was adopted, taking \ less than - but not so 5 small as j, we have for the approximate value of this ratio to the order of accuracy necessary: to this order, J. A in fact, Maclaurin's result agrees with that which we obtained. Now with the hypothesis of fluidity we may find the ratio by the aid of Clairaut's theorem ; for the ratio of the difference of the gravities to the half-sum is the same as the ratio of the differ- ence of the gravities to the equatorial gravity, to our order of accuracy. Thus, by Art. 171, the ratio is 5; * 5 ; 22' that is 2 "Z 2 X Now this is not necessarily less than the former value ; it is in fact greater if X is less than ^- . 265. Maclaurin considers in his Articles 668... 671 the attrac- tion of an ellipsoid of revolution made up of similar and concentric shells of varying density. He shews theoretically how to deter- mine the attraction on a particle on the axis, or in the plane of the equator, either external or internal. In modern language we should say that he reduces the general problem to depend on a single integration : see Art. 261. Maclaurin then takes special 158 MACLAURIN. cases ; he treats briefly the case in which the density varies inversely as the diameter of the shell, and the case in which it varies inversely as the square of the diameter ; and more fully the case in which it varies as the diameter. Jacobi has made an important remark on the subject of the similar concentric shells when the ellipsoid is not of revo- lution : see Poggendorff's Annalen, Yol. xxxm. 1834, page 233. Pontdcoulant, Theorie Analytique, Supplement au Livre V. page 22. 266. It will be interesting to discuss analytically some cases of similar concentric shells with varying density. I. Suppose the density to vary inversely as the diameter. Put x for ea ; then the density varies inversely as x ; say that the density = - . Take the formulas of Art. 261, omitting the MP common factor 3 -- - ; thus we find that the attractions for an external particle in the plane of the equator and on the axis respectively are xdx , xdx 8 r being the distance of the particle from the common centre of the shells. The limits of integration are and ce, where c is the semi-axis major of the bounding shell of the solid. Thus we obtain respectively and i log Suppose now that we take the external particle to be on the surface of the oblatum ; then in the former expression we put r = c, and in the latter we put r 2 = c 2 (l e 2 ). In both' cases we obtain a result independent of c. Thus the attractions at the equator and at the pole are independent of the size of the obla- tum. Maclaurin gives this result so far as relates to the attrac- tion at the equator. It is also true that for a particle situated at any point of the surface, the attraction will be independent of c; this may be shewn by reasoning of the kind given in Art. 242. MACLAURIN. 159 II. Suppose the density to vary inversely as the square of the diameter. In this case we find, omitting the same common factor as before, that the attractions for a particle in the plane of the equator and on the axis are respectively dx , f dx . .,_ a . and I j 1 . _! ce , 1 , _ t ce that is, - sin - - and - tan - . Suppose now we take the external particle to be on the sur- face of the oblatum ; then in the former expression we put r = c, and in the latter r = c^(\ e*). Hence we see, that for oblata similar in form but different in size, each result varies inversely as c. Maclaurin gives this result so far as relates to the attrac- tion at the equator. It is also true that for a particle situated at any point of the surface the attraction will vary inversely as c ; this may be shewn by reasoning of the kind given in Art. 242. /> Also, since sin" 1 e = tan" 1 -rp. - ^ , we see that for the same V\* ~ e J ellipsoid, the equatorial and polar attractions for a particle on the surface are inversely as the equatorial and polar diameters. Maclaurin does not mention this. I add, that the law of density under consideration is the only law which gives the result just obtained ; the density being assumed to be a function of the diameter of the shell. To prove this : assume the law of density to be represented by ^-^ . Then we require that / ,, , X a; 2 Jo cVc x should be to r *fe) *** ^ c ^(1 -e 2 ) is to c. Jo r cV-far Assume in the first integral x = c sin 0, and in the second x = c V(l e 2 ) tan 6 : then we arrive at rsin" 1 e [(/> (c sin 6} - <f> {c V(l - e 2 ) tan 0}] dO = 0. J o 160 MACLAURIN. Differentiate with respect to c and to e ; thus /sin" 1 e 0=1 [sin 0<f> (c sin 0) - V (1 - e z ] tan 0$ {c V(l - O tan 0}] d0, rsin" 1 e />/> and 0=1 .., ,. tan 6$ (c V(l - e 2 ) tan 0} d0. Jo v(*-~~0) Multiply the latter by - , and add to the former ; thus CG we obtain r sin" 1 e sin #</>' (c sin 0) c?0 = ; J o and by differentiating with respect to e we see that c//(ce) = 0. This shews that <(#) must be a constant. III. Suppose the density to vary as the diameter. In this case, omitting the same common factor as before, the attractions for a particle in the plane of the equator and on the axis are respectively x 3 dx f x*dx Thus we shall obtain when the external particle is on the surface |{2-3V(l-e ! ) + (l-e s ) 1 ) and | {/+ (I -e 2 ) log (1 - e*)}. Each varies directly as the square of c. And, as before, for a particle situated at any point of the surface the attraction will vary as the square of c. In this case the ratio of the equatorial attraction to the polar is 2 2- 3 e Expanding in powers of e 2 we shall find that this becomes MACLAURIN. 161 thus if we neglect the square and higher powers of e a , the two attractions are equal. This agrees with a statement in Maclaurin's Article 673. IV. Suppose the density to vary as the cube of the diameter. In this case, omitting the same common factor as before, the attractions for a particle in the plane of the equator and on the axis are respectively x 5 dx x s dx Thus we obtain when the external particle is on the surface J (8 - 15 (1 - erf + 10(1 - erf - 3 (1 - erf}, and ~ [e 4 - 2e 2 (1 - e 2 ) - 2 (1 - e 2 ) 2 log (1 - e 2 )]. Each varies as the fourth power of c. And, as before, the same result will hold for a particle situated at any point of the surface. The ratio of the former to the latter when we neglect the + square and higher powers of e" is 267. Maclaurin in his Articles 672 and 673 supposes that his shells are fluid, and that the density varies as the diameter. He comes to the conclusion that the ellipticity is rather greater than it is for the case of uniform density; but that the increase of gravity in passing from the equator to the pole is less than for the case of uniform density. He also briefly states the results for the case in which the density varies as the cube of the diameter. The results are of no value, for Maclaurin merely assumes Newton's principle of columns of fluid balancing at the centre, and does not shew that the whole fluid will be in equilibrium. In fact it is known that the whole fluid will not be in equilibrium. If the density of the shells varies the excentricity can not be constant. The objection to Maclaurin's investigations was noticed by Clairaut : see his Figure de la Terre, pages- 229... 232. T. M. A. 11 162 MACLAURIN. For an example we will give the investigation, on Maclaurin's principles, of the case in which the density varies as the cube of the diameter. Denote the attraction for a particle on the surface at the equator by E, and at the pole by P, the density at the surface by p, and the centrifugal force at the equator by V: let 2a and 25 be the equatorial and polar diameters. For the equatorial column, at a distance x from the centre, x* x the attraction is E ? , the centrifugal force is V- , and the den- x 3 sity is p 3 : hence the weight of the column fVi^ 4 T7 oA x* , (E V\ = [E-r - F- o-3-cfo= - pa. Jo V a a/ r a \S 5 / r p Similarly the weight of the polar column = pb. /E V\ P Therefore f g- - -gj pa = -^ p&. We take from observation _ Therefore -^ = ^ ~o" = - 5 - approximately. * 1 1 5 x 289 5 x 289 ~F (P But we saw in Art. 266 that = 1 + ^ nearly; z o therefore - | = (l - ^^) (l + ~] ; 8 1 _ ~5 X 5 X 289~226' The ratio of the polar gravity to the equatorial P 1 1 e 2 13 MACLAURIN. 163 Thus we obtain an excentricity slightly greater than for the 02 -^ case of uniform density, where ^=- ; but the increase of L ZoO gravity in passing from the equator to the pole is much less than for the case of uniform density, where it is x of the whole. 268. Maclaurin devotes his Articles 674... 678 to the discus- sion of the case in which the density involves two terms, one con- stant, and the other varying as the diameter of the shells. Let x represent the diameter of any shell, a the diameter of the out- side shell ; then he takes the density to vary as -- ^ ~~ x - '^ 1 * s obviously amounts to supposing the density to vary as the distance from some point beyond the outside shell. Maclaurin's discussion of the attractions at the equator and at the pole is very clear and satisfactoiy. Assuming as before that the body is fluid, and using Newton's principle of columns balancing at the centre, Maclaurin arrives at the following results : If e and j have their usual meanings j r _ 3 (3n + 1) (n - 1)] \ 17w* + 34ra + 45 J * 4 VJn* +34n + 45 8 The ratio of the difference of polar and equatorial gravities to their half-sum is 5j J 8(n + 8)(fi-l)] 4 \ r 17w 2 + 34rc+45j* Maclaurin says in his Art. 678, ...no supposition of this kind can account for a greater variation from the spherical figure, and at the same time for a greater increase of gravitation from the equator to the poles. . . . If we put n in the above value of e* we get e 2 = -- ; the 3 density now varies as the diameter : the result coincides with that obtained by Maclaurin in his Art. 673. Maclaurin in his Article 679 states the results obtained by substituting for n in the above general formulae, the values 2, 3 and infinity. 112 164 MACLAURIN. 26lf. Problems of the kind considered by Maclaurin in his Articles 672 . . . 679 had previously engaged the attention of Clairaut: see Chapter VI. Both Clairaut and Maclaurin how- ever failed, from not knowing that the equilibrium of the whole fluid was impossible on their hypotheses. Considered merely with respect to attractions both supplied interesting results: Clairaut gave approximate values of the attraction at any point of the surface, and Maclaurin gave exact values of the polar and equatorial attractions. The failure as regards the hydrostatical part of the problems was recognised by Clairaut himself: see his Figure de la Terre, pages 155 and 259. 270. Maclaurin in his Article 680 takes the case of an oblatum which is composed of shells of finite thickness ; each shell is of uniform density, but the density varies from shell to shell, in- creasing towards the centre : the bounding surfaces of the shells are supposed to be all similar and concentric. He gives, in fact, an approximate expression for the excentricity in the case of one shell surrounding a central portion, from which it appears that the excentricity is less than for the case of a homogeneous fluid ; and he states that a similar result will hold when there are more shells. Let us investigate the general result which is briefly indicated in Maclaurin's Article 680. First, let there be one shell surrounding a central part. Denote the density of the shell by 1, and that of the central part by 1 + tr. 2a Let the equatorial diameter of the central part be , where 2a is the outer equatorial diameter of the shell. We proceed with Maclaurin to equate the weights of the equatorial and polar columns. We begin with finding the weight of the equatorial column. Let x denote a distance from the centre, 7 the density at this point, </> (x) the attraction at this point. Then the weight of the ra column will be denoted by I 7$ (x) dx ; and we must observe that Jo 7 and </> (x) have different forms at different points. MACLAURIX. 1G5 Put k for - \/(l e 2 ). Then, from x to # = - we have l + cr, and </> (#) = k(i + cr) ( 1 + TTT) * 5 an d from a; = - to Here we only retain the first power of e 2 ; and this we shall do throughout the investigation. See Art. 261. fa Hence we shall find that I <y$(x}dx becomes Jo If F denote the centrifugal force at the equator, the effect of the centrifugal force on the column is Vl 1 + -3 J ~ . We put as usual y = ?*; for the denominator on the left-hand side ex- presses the attraction at the equator to the order which we are here considering. Thus the effect of the centrifugal force on the column is jk ( 1 + 3 j ( 1 + 3 J -^ . In a similar manner we find that if 2& be the outer polar diameter of the shell the weight of the polar column is denoted by , fn-1 a e*(n*-l) a 5 } + k<r\ 3 . j- -- ^-5 -. jz\-a\ [ n* b 5?i 5 b 3 } The factor 1 1 ^J maybe obtained thus: the attraction at the pole of an oblatum of density unity is k ( j =- . ^ J, that is, 100 MACLAUIUN. kb f I -f - ) nearlv; thus the weight of tin- polar column, if tlio density were unity throughout, would be k o (l-f ' ), that is, Equate the weights of the columns; thus we get there f o re f'V 2 + ?i V + iC(j 4- -. (/r \} this is less than -^ ; - , since cr is positive and ?i greater than unity. Maclaurin gives this result. Let us now suppose that there are three portions of fluid, an outer shell of density 1, a second shell of density 1 + p, and an inner part of density 1 + p -f a. Let the equatorial diameter of the inner part be ; and Jet the outer equatorial diameter of the 'In second shell be . It is easy to see that the value of c 2 wi m now be determined bv an equation of the form fy ft _, _p_ +~Vi + p .,+ in" >r J \ in" H' 1 4-tL-rms ot the first and second degree in p and Now with re-pert to the denominator on the right-hand side, we know t hat if p it reduces to rr- / I 1 \ .'} >r - 1 1 + , -M : + -. and if </ U it will jvdu<v <" ;J similar e.\prc.^-ion in ^ and m. MACLAURIN. 167 Hence, in fact, we have only the term in pa- to find. Proceed as before : we see that in estimating the weight of the equatorial column we have a term Se" * 3 3 To e"\ a? o) 2n< and in estimating the weight of the polar column we have a term This shews that the term we are seeking is 2 5 n-w 3 n s -m*\ . (I The term which involves pa- in the numerator is 50 H 5 i , ra V ra V which is certainly less than the term which involves pa- in the denominator. There will no difficulty in extending this. Suppose that there are four portions of fluid, and that their densities are 1, 1 + tsr, P, l+'cr + p + cr; and the corresponding equatorial semi- diameters a, j , , . Then the numerator of e 2 will now be I m n / fl4- -4- r 4- I A * J2 ~ i 2 The terms in the denominator can easily be written down ; that in par is the same as before ; that in -crp will in like manner 1 I SI (?7i 2 - Z 2 ) 2m 5 m OT " W The problem is of no importance ; for, as we have said, the whole fluid mass will not be in equilibrium : but still there is something curious in the simplicity of the solution when con- sidered with regard to the complexity of the hypothesis. 271. Maclaurin in his Article 681 takes the following hypo- thesis : let there be a shell of fluid, the bounding surfaces of which are concentric and similar oblata; and within the inner surface let there be a solid concentric sphere. He again equates the 1C8 MACLAURIN. weights of the equatorial and polar columns of fluid. It is obvious that the hypothesis is not consistent with the known conditions for fluid equilibrium, unless he supposes the inner surface of the fluid to become rigid; and if this is supposed, the weights of the columns will not be equal. Clairaut pointed out that the hypothesis is untenable : see his Figure de la Terre, page 256. We will state the results which will be obtained on Maclaurin's principles. Take the density of the solid and of the fluid to be the same, and uniform : let 2a and be the external and inter- nal equatorial diameters of the shell. Suppose the volume of the sphere to be -^ we shall obtain sphere to be - of the volume of the oblatum if complete ; then Maclaurin's result agrees with this ; but he uses the word area for volume. The ratio of the difference of the polar and equatorial gravity to the semi-sum will be found to be Maclaurin has n*N where we have -^ , and he has where we have lOn*. We may verify by putting N infinite and n = 1 ; then we have only an indefinitely thin shell, and we get e* = 2j : and the excess of polar over equatorial gravity becomes zero by our formula, as it should. If we put n = 2, we find that Maclaurin's result would in general be negative, supposing we make the correction for J\T. Maclaurin next supposes that the central part instead of being, a sphere is an ellipsoid of revolution ; he gives the correct result on his principles, supposing the ellipticity of the central part to be MACLAUKIN. 169 small : he has not formally stated this condition, though he has certainly used it. The following is his result: let the distance from the centre to a focus of the inner part be - ; then the rest of of the notation being as before, Suppose, for example, that the surface of the solid part coin- 1 e cides with the inner surface of the fluid, so that - = - , and N=n 3 : then we obtain e 2 = ^ > as it should be. Maclaurin goes on to say that other suppositions might be made, but implies that it is not desirable to dwell on them. He makes the following very judicious remark : When more degrees shall be measured accurately on the meridian, and the increase of gravitation from the equator towards the poles de- termined by a series pf many exact observations, the various hypotheses, that may be imagined concerning the internal constitution of the earth, may be examined with more certainty. 272. Maclaurin gives in his Articles 682... 685 some remarks on the shape of the planet Jupiter. Suppose a satellite to describe round its primary in the plane of the primary's equator, a circle of radius r in time T\ let the primary revolve on its axis in time t ; let a and a V(l e 2 ) be the r 3 t z semi-axes of the primary. Maclaurin puts ^V for -3 x -^ . (.1 To connect N with j and e we have the following equations : see Art. 261 : 170 MACLAURIN. + JQ]F l+ 565* + "" 1 " t "Io" f 56" + "" therefore Nj = v where M stands for - . a Put for j its value from Art. 262 ; thus Now Maclaurin says in his Article 660, that "the excess of the semidiameter of the equator above the semiaxis is to the mean 10 3 semidiameter as 5 to 4<N+ -=- - -*rrr nearly;" and in his Arti- cle 682 he says, "By continuing the series in art. 660 one step further, the excess of the semidiameter of the equator above the semiaxis is to the mean semidiameter as 5 is to -=- -. + > " ^ et us examine the last state- ment. We have just seen that W- We can infer from Maclaurin's result that he rejects the e* squares of -^ ; and, indeed, if we look at his numerical values, it will appear that to the order he considers, he might have e 2 e 2 rejected - also. However, retaining - , we have from (1), 5 2N l+l~- " !- - 25 ~< (2). MACLAURIN. 171 For a first approximation we have from (2) *_ 5 ~23T Substitute this value in the denominator of (2), neglecting e*, then for a second approximation A A 2JV 2N this agrees with what Maclaurin gives at the beginning of his Article 660. For a third approximation we substitute for e 2 in the deno- minator of (2) the value -^. ( 1 oTr^) 5 an d so we get 5 6 2 = 3 /5 /25 SIX 'AZW^Vw ^6 196/ 4.", 3 25 x 13 H 283T 4iYJf 2 + 8x 2 Now we require the value of j; and this is 1 2 45 3 25 x!3 5 h + 8 x 196-tf ' 4iY that is, - 5 10__3^ 25 x 13 ' 7 M* + 2 x 196JV 4825 325 Thus instead of Maclaurin's large coefficient - Q J1-- we get only -r^ . I/- MACLAURIN. -7*>. Maclaurin tiiuls that his calculation brings out too great an ellipticity for Jupiter, making the lunger diameter to he the shorter, about as !():> to IKS; whereas, according to Cassini, the difference of the diameters was about of the longer diameter, and according to Pound between and - -. Maclaurin then makes the supposition which we have noticed in Art. 2(58, that the density varies as - _^ ^', he gives the general result, and putting ?z=-i in this, he finds a tolerable agreement with ob- servation. But I am unable to verify his general result. By the aid of the expression given in Art. -01 for the attraction of a shvll on a particle in the plane of the equator, I obtain with the notation of Art. '27'2, e 2 - n + :> 4- - (n + 5) . Maclaurin's result in this notation is sr if we multiply both numerator and denominator of the last frac- tion by 1 -_- , and neglect c'\ we get n + > + ! (?t + .">) Maelaurin cannot be correct ; loi- it is certain that if J/ 1 we oU"ht to lia \ '< A7 1 . o *y 1^71. Some otlici' investigations respecting attractions are contained in Articles !)00... !)().> of Maclaurin's Fluxions. MACLAURIN. 173 Here he supposes the law of attraction to be that of the nth power of the distance ; he says that n is to be less than 3 : it will be found on examination that he means n to be algebraically less than 3, and does not assume 'n to be necessarily an integer, so that in fact 3 n must be positive. Maclaurin considers the attraction of an ellipsoid of revolution on a particle at the equator or at the pole; as we should say in modern language he reduces the problem to a single integration. He says in his Article 904 as his general conclusion, "Hence, therefore, the gravity at the equator, as well as the gravity at the poles, is measured by circular arks or logarithms when n is any integer number less than + 3." Maclaurin refers in his Article 905 to " a late ingenious essay, Phil Trans. N. 449. by Mr Clairaut:" see Art. 167. 275. We will now notice the bearing on our subject of Maclaurin's Prize Essay on the Tides, which was mentioned in Art. 232. Maclaurin in Lemma III. of his Essay gives matter equiva- lent to Articles 628... 630 of the Fluxions: see Arts. 242 and 243. In Lemma IY. he gives matter equivalent to Articles 631... 634 of the Fluxions : see Art. 244. The Propositio I. Theorema Funda- mentale of the Essay contains the important results enunciated in Article 636 and demonstrated in the following three Articles of the Fluxions ; see Art. 245. Maclaurin briefly indicates the application of this fundamental theorem to the Figure of the Earth, supposing that the Earth is a fluid of uniform density ; the theorem gives the ratio of the axes, and the direction of gravity at any point. He says : " Hsec omnia accurate demon- strantur ex hac Propositione ; qua? quamvis in disquisitione de figura Terra? eximii usus sint, hie obiter tantum monere con- es venit." Lemma V. of the Essay corresponds to Article 642 of the Fluxions; though it is rather less general: see Art. 250. By means of this Lemma the calculation of the attraction of a solid of revolution on a particle at its pole is made to depend on rinding the area of a certain curve. 174- MACLAURIN. Propositio II. of the Essay determines the attraction of an oblongum on a particle at its pole; tin- method is substantially the same 1 as that in Article 647 <>f the Fluxions, but in the Essay the notation is that of the Differential and Integral Calculus, not that of Fluxions and Fluents: see Art. 252. At the end of the proposition Maelaurin briefly indicates the result for the case of an oblatum; this case is worked out in Article 64-6 of the Fluxions. For the subject of the Tides the ol>luit<juni is the important figure, while for the subject of the Figure of the Earth the oblatum is the important figure. In Lemma VI. and Proposition III. of the Essay, Maelaurin estimates the attraction of an oblongum on a particle at the equator, and briefly indicates the result for an oblatum ; the method is substantially the same as in Articles 640 and 647 of the Fluxions. Thus we see that at the date of the Fssay on the Tides Maelaurin had completely solved the problem of the attraction of a homogeneous ellipsoid of revolution on an internal particle. The Treatise of Fluxions contains in addition the theorem respect- in^ the attraction on an external particle which we have noticed O I iii Art. 25i) ; and also the propositions respecting ellipsoids of revolution, not homogeneous, which we have noticed in Arts. 256, 264 and 265. 276. Maelaurin died in 1746, so that he survived the pub- lication of Clairaut's Fiyure de la Terre. It does not however appear that he published anything on our subject after his Fluxions. In the last year of his life he was obliged to leave his home in consequence of the rebellion in favour of the Stuarts; and the hardships he thus encountered seem to have laid the foundation of his mortal illness: in the premature death of the most famous of her sons Scotland paid a heavy price for the temporary success of the Pretender's enterprise. The importance of Maclaurin's investigations may be seen by observing how great h.'is been his influence on succeeding writers. Clairaut, DAlembort, Lagrange, Logvndro, Laplace, Gauss, Ivory and Chasles shew by reference explicit or implicit MACLAURIN. 175 their obligations to the creator of the theory of the attraction of ellipsoids. Maclaurin well deserves the memorable association of his name with that of the great master in the inscription which records that he was appointed professor of mathematics at Edin- burgh, ipso Newtono suadente. In the application of the theory of Attraction to the Figure of the Earth Maclaurin was impeded by the imperfect state at that time of the knowledge of the conditions of fluid equilibrium, and also by the want of accurate measurements; the latter cir- cumstance led him to suppose that the ellipticity was greater than it really is. Nevertheless he was the first to demonstrate exactly the possibility of the relative equilibrium of an oblatum of rotating fluid. See Art. 249. CHAPTER X. THOMAS SIMPSON. 277. THOMAS SIMPSON published in 1743 a volume entitled Mathematical Dissertations on a variety of Physical and Analytical Subjects. The volume is in quarto ; the Title, Dedication, and Preface occupy viii pages, and the text occupies 168 pages. 278. The first essay extends over 30 pages; it is entitled A Mathematical Dissertation on the Figure of the Earth. In the preface Simpson speaks of this as " one of the most considerable Papers in the whole Work,..." ; and after referring to the contents of the essay he says : ... I must own that, since my first drawing up this Paper, the World has been obliged with something very curious on this Head, by that celebrated Mathematician Mr. Mac-Laurin, in which many of the same Things are demonstrated. But what I here offer was read before the Royal Society, and the greater Part of this Work printed off, many Months before the Publication of that Gentleman's Book ; for which Reason I shall think myself secure from any Imputations of Plagiarism, especially as there is not the least Likeness between our two Methods. In a foot-note he says It was read before the Royal-Society in March or April, 1741, and had been printed in the Philosophical Transactions, had not I desired the contrary. The preceding extract might seem to establish for Simpson the priority over Maclaurin in the first enunciation of some of the most important results in our subjects ; but Simpson makes no THOMAS SIMPSON. 177 reference to Maclaurin's prize Essay on the Tides which belongs to an earlier date than March, 174-1, and contains the essence of much that was expanded in the Treatise of Fluxions: see Art. 275. Thus Maclaurin's claims remain indisputable ; but as we shall shew there are some very important points in which Simpson had no predecessor. Simpson's essay is very remarkable, as we shall see by an analysis of its contents. 279. The first fourteen pages bring out exact expressions for the attraction of an oblatum on a particle at the surface ; Maclaurin as we have seen had previously effected as much. The following is the essential part of Simpson's method : suppose an ellipse to revolve round a tangent at one end of an axis, through an indefinitely small angle ; a wedge-shaped element is thus produced, and Simpson calculates the attraction which this ele- ment exerts on a particle placed at the point of tangency. The whole oblatum is cut up into such wedge-shaped elements, and so the resultant attraction is determined. Instead of the elegant geometry of Maclaurin, Simpson employs analysis, the style of which for its rude strength reminds the reader of that of Laplace. 280. In the course of his investigation on his page 3, Simpson has in effect to determine the value of -77= r I . , in the form of a series proceeding according to ascending powers of #. He expands the expression under the integral sign in powers of g, and effects the integration ; then he multiplies this by the expan- sion of j-(] \ an d arranges the product. He does not however demonstrate the form of the general term, but seems to assume it from observation of a few simple cases. As all his subsequent investigations rest on this, it seems strange that he did not proceed here with rigid exactness. We may of course obtain the required result easily in another way. Assume x = =- 5-3 ; thus we find that the integral is 1 + g cos' 6 ' , f ^ sin 2 cos 0^0 transformed into 2 T. M.A. 12 178 THOMAS SIMPSON. Then, expanding in powers of g, we see that the general term = 2 (n + 1) ( - l)y [ ** (1 - cos 2 0) cos 2n+1 6 dO J o 2w 4- 2 " ... k )9 3. 5. 7. ..(271+3)' This agrees with Simpson's result. 281. The preceding Article furnishes the only instance of an imperfect investigation which I have noticed in Simpson's essay : there are however, as might be expected, cases in which his processes may be simplified. Perhaps the most important part of his analysis consists of the evaluation, on his page 10, of the following definite integrals : r\T { (a cos e + A sin 6}** - (a cos - A sin <9) 2w } sin cos dO, J o rfr and {(a cos 4- A sin 0}*" + (a cos 6 - A sin &f n ] sin 2 B dO. J o We will consider the second of these.; our remarks will be easily applicable to the first. Simpson expands (a cos 6 -f A sin 0) 2 * and (a cos 6 A sin #) 2 *, and then integrates each term separately; the following is a simpler method. It is obvious that if we were to expand, our final expression would involve only even powers of sin 6 and cos 6 ; and thus we may use and 2?r as the limits of integration, and take one fourth of the result. Assume a = 7c cos ft and A = k sin fi ; so that & 2 = a 2 + A 2 ; then the definite integral becomes \ (a 2 + A*} r {cos 2n (0 - /3) + cos'* (0 + /3)J sin 2 dO. J o Consider f " cos 2n (0 - 0) sin 2 d0. THOMAS SIMPSON. 179 Put sin 6 = sin (0 + 6 - &) = sin {3 cos (0-13) + cos sin (0 - 0); Thus we get cos 2 * ((9 - /S) (sin cos (0 - ) + cos sin (0 - )} 2 <J0. Put < for ; then the limits of the integration for </> are /3 and 2?r . The integral Icos 2 " +1 </)sin <f>d<f> is zero between these limits ; so that we are left with ~ ft (sin 2 /? cos 21 "* The limits may be changed to and 2?r, because the expression to be integrated has the same value when (p 2?r a as when <f) = a. T27T Transform I cos 2w (6 + /3) sin 2 c?^ in a similar manner. Jo Thus finally we obtain (cos 2n (0-ff)+ cos 2 " (0 + /S)} sin 2 (9 cZ0 = 2 cos 2 f 2 " cos 2 * <bd6+2 (sin 2 - cos 2 8) (^ cos 2n+2 <f> J<f> ; Jo Jo we have now a well-known definite integral form. 282. It should however be observed that Simpson's series are not always convergent. For example, on his page 13 he has the series which results from expanding tan" 1 ^/^ in powers of *JB, and B is not necessarily less than unity. 283. Having obtained accurate expressions for the attraction of an oblatum on a particle at the surface, Simpson considers the relative equilibrium of a mass of rotating fluid. He says on his page 16, "the Form which that Fluid must be under, to preserve this Equilibrium of its Parts, is that of an oblate Spheroid." It is almost needless to remark that Simpson does not demonstrate this ; he demonstrates that the figure which he assigns is a possible 122 UNIVERSITY OF n*. 180 THOMAS SIMPSON. figure of relative equilibrium, and not that it is the only figure : see Art. 168. Simpson contents himself with shewing that Huygens's con- dition for fluid equilibrium is satisfied. Laplace gives, in the Mecanique Celeste, Livre in. 20, the following equation which connects the excentricity of the oblatum, supposed small, with the angular velocity Simpson gives this on his page 19 in his own notation, and 125 x 37 supplies the third term of the series, namely - ^- q s in Laplace's notation : Simpson remarks that this is very nearly the same as 35 " 14 - 30? > and the approximation will be found extremely close as far as q s . Simpson on his pages 15 and 20 demonstrates the truth of some approximations given by Stirling ; see Chapter V. 284. We now arrive at the most important part of the Essay. Simpson shews, to use modern language, that if the angular velocity of rotation exceeds a certain limit, the oblatum is no longer a possible form of equilibrium. This proposition has since been incorporated in the Mecanique Celeste, without any reference to Simpson : see Livre in. 20. Laplace uses \/(l + X 2 ) to express the ratio of the major axis to the minor axis in the oblatum, and Simpson uses V(l + x 2 ) ; for the extreme case in which equilibrium is possible, Simpson gives x = 2-5293, while Laplace gives X = 2'5292. Ponte*coulant agrees with Laplace ; see his Theorie Analytique. . ., Vol. ii. page 399. Poisson agrees with Simpson ; see his Me'canique, Yol. ii, page 542 : so also does Resal ; see his Traite JElementaire fo Mecanique Celeste, page 196. Simpson's investigation, though less elaborate than Laplace's, is adequate and satisfactory. THOMAS SIMPSON. 181 285. For any angular velocity less than the limit to which we have alluded in the preceding Article, there are two and only two possible oblata; this has been shewn by Laplace in the section already cited. According to Laplace, D'Alembert first observed that more than one figure of equilibrium might correspond to the same angular velocity without however determining the number of such figures: see Laplace's Figure des Planetes, page 124, and the M&anique Celeste, Livre XL 1. Ivory makes a similar remark in the Philosophical Transactions, 1834, page 513. But it should be observed that although D'Alembert may have first explicitly pub- lished the statement, yet Simpson gives a Table which distinctly implies the fact. The Table in substance is the following : 1 to 1-01 11-236 08925 1 to 1-05 5137 1978 1 to 1-5 2-056 5568 1 to 2 1-814 6944 1 to 27198 1-7226 8105 1 to 4 1-810 8774 1 to 7'57 2-118 92705 1 to 10 2-338 9216 1 to 20 3110 8728 1 to 40 4-275 8000 1 to 100 6-600 7033 1 to 1000 20-640 4845 This Table is given on Simpson's page 24, with the exception of two lines which I have supplied from other parts of the essay. The first column expresses the ratio of the minor axis to the major axis of the revolving oblatum ; in Laplace's notation it is -T-27 . The second column is Laplace's - r ; thus it is in- versely proportional to the angular velocity, and so directly pro- portional to the time of rotation ; it may be considered to express the time of rotation if we take a certain unit of time, the unit being the time in which a satellite would revolve round a sphere equal in volume and density to the oblatum, moving close to the 182 THOMAS SIMPSON. surface : this is Simpson's own interpretation. The third column we will speak of presently. An inspection of this Table shews that in the second column the figures decrease down to some minimum, and then increase again : thus it is obvious that corresponding to an assigned angular velocity there are in general two values of V(l + X 2 ). 286. Let us now explain the third column of the Table. Simpson uses the term momentum of rotation for the sum of the products of the mass of every particle into its velocity. Let co be the angular velocity, 2a the major axis, 2b the minor axis, p the density ; then it is easy to shew that the momentum of rotation of 7T* the oblatum is -j pfoa 3 b. Now suppose a sphere, equal in density 4 and volume to the oblatum, rotating in the unit of time specified in Art. 285. The momentum of rotation for the sphere would be 2 2 4 , where R 3 = a?b ; so that it would be -r pa) l (a*b)*. The ratio of the former value to the latter is therefore that is ( y j , >! \bj I j (1 + X 2 )^. But Simpson has taken the unit of time so that q l = 1 ; hence the ratio becomes <?"*(! -f X 2 )*. Thus the third column can be obtained from the first and the second ; we must divide the cube root of (1 + X 2 )^ which is given in the first column by 7= which is given in the second column. Simpson's third column has not any physical interpretation, though he himself by mistake supposed that it had. For he uses the term quantity of motion on his page 21 in the same sense as angular momentum ; and he erroneously says that it " will be no ways affected by the Action of the Particles upon one another while the Figure of the Fluid is changing." Then on his page 22 he gives a discussion as to the greatest possible value of the quantity of motion for a given mass. What he must have intended to employ is the principle which in modern language we call the Conservation of Areas. This is THOMAS SIMPSON. 183 plain from what he says in a note on page 135 of his Miscellaneous Tracts, 1757; here he admits the mistake in the present work. Instead of the sum of the products of the mass of every particle into its velocity, he should have considered the sum of the products of the mass of every particle into what we may call its areal velocity. Laplace uses this sum in the Mecanique Ce'leste, Livre ill. 21. He there has an equation </> = 0, which agrees substantially with one given by Simpson on page 136 of his Miscellaneous Tracts. Simpson however does not discuss the equation ; Laplace shews that it has only one solution. 287. In the Table of Art. 285, the fifth line and the seventh line are not inserted by Simpson, though he has supplied the materials for them in the course of his essay. In the fifth line the entry in the second column gives the minimum value of that column ; it really occurs in page 20 of Simpson's essay in the form -^yr^ , so that '58053 is the value ' of V# which corresponds to Laplace's value of '337007 for q. The corresponding number in the third column by Art. 286 is therefore (27198)* x -58053. In the seventh line the entry in the third column gives the maximum value of that column ; Simpson finds on his page 22 that for this case X = 7'5 nearly. The corresponding number in the second column by Art. 286 is therefore (7'57) * -r '92705. 288. Simpson shews on his page 22 that the gravity at any point of the surface of the oblatum varies as the length of the normal between the point and the axis of revolution. See Arts. 153 and 247. 289. Simpson having thus discussed the case of a homo- geneous oblatum, proceeds to the case in which the oblatum is not homogeneous. He supposes that the oblatum consists of a central portion which is spherical and denser than the rest, and of an outer portion ; each portion is supposed homogeneous. 184 THoMAS SIMPSON. If we change the sign of X in a result which was obtained in Art. 2(J4-. page loii, we have i + | and Simpson's result agrees with this. Simpson does not shew that his fluid mass will remain in equilibrium; he contents himself with making the resultant force at the surface normal to the surface: if we suppose his central portion to be solid, the conditions of equilibrium will be satisfied. With the exception of this defect, Simpson's investigation of the value of the ellipticity and of the variation of gravity along the surface is quite satisfactory. In finding a definite value for the ellipticity, Simpson gives a better treatment of the problem than Maclaurin did in his Articles (Kjil and (>(>7. Simpson briefly applies his result to the case of the planet Jupiter, lie concludes thus: ... but as no Hypothesis, for the Law of Variation of Densitv, can (from the Nature of the Thing) he verified either hv Experiments, made on Pendulums in different Latitudes, or an actual Mensuration of the Degrees of the Meridian, I .shall in.-ist no further on this Matter, hut content mvself with having proved in general, that the greater the Density is towards the Centre, the less will the Planet differ from a Sphere, and the greater will be the Variation of (Jravitation at its Surface. 2!)(). The second essay in Simpson's Mathematical Diwrta- tionx is contained in pages .'>!.. .,*>7; it is entitled A General Tm'efiti'yation <>f the Attraction at the, ft ur face* <>/' /todies nearly .,,/,,,/;./. Tin- essay b-gins with investigating the attraction of a wedge- shaped element like that in Art. L'7'-' u a particle in a certain position; the boundary however is now not an ellip>e lnit any curve which is nearlv circular. Take tor the equation to this boundary THOMAS SIMPSON. 185 where 5 2 , b s , 6 4 ,... are supposed to be so small that their squares and products may be neglected ; the boundary passes through the origin : suppose that it cuts the axis of x again at the point for which x = a. Let the figure revolve round the axis of y through an infinitesimal angle Scf> ; then the attraction of the element generated by the revolution of the area 2ySx on a particle at the origin, resolved along the axis of x is x(f> -rr^ - jr . Hence the x v (x + y ) r at 11 dx attraction of the whole wedge-shaped element is 2S< I ~ - ^ . ; o v \ x + y ) As in Art. 280, Simpson gives the correct value of this integral ; but he does not strictly demonstrate his result. We will supply the demonstration f a ydx [ a *J(cx - x* + If? + b,x 3 + b 4 x 4 +...), J o V (x* + /) ~ J o J(cx + bjf+ bj? + bjt + . . .) Now by supposition c = a b z a b 3 a? b 4 a 3 . . . ; substitute this value of c in the expression under the integral sign, divide both numerator and denominator by fjx, and expand. Hence we find that the above integral becomes where n is to have all positive integral values beginning with 2. To effect the integration put x = a sin 2 6 ; thus we get f T ( 2aj sin^cos 2 ^ 1-... that is, r'amO J2acos 2 l9- ... - b n cT l (1 - sin 2 "- 2 ^) sin 2 l9- ...\d0. Jo I ) Thus finally we obtain for the attraction required I ^ sin J2a cos 2 - (\a + b,a? + b.a 3 + . . .) sin 2 01 d0 ft* 4- a series whose general term is I b n a n 1 sin 2n+1 6 dd. JO 186 THOMAS SIMPSON. Then making use of the value of c, we find that this becomes 2 ..2.4... 2ra K c + a series whose general term is b n a ^ -r- . In the small terms we may put c for a, so that our result is -c + ^bc 2 ' 4 - 6 5c 8 2.4.6.8^^3 This agrees with Simpson's result. t 291. Having thus obtained the attraction of the wedge-shaped element, Simpson proceeds to the attraction of any solid of revolution which is nearly spherical : his final result on his page 37 gives the expressions for the resolved attractions, along the normal and along the meridian tangent, which such a body produces on a particle at its surface. 292. The pages 41... 45 of Simpson's Mathematical Disserta- tions contain an essay entitled To determine the Length of a Degree of the Meridian, and the meridional Parts answering to any given Latitude, according to the true spherodical Figure of the Earth. This essay gives an approximate expression for the length of a degree of the meridian, on the hypothesis that the earth is an oblatum ; a small Table is supplied of the length of a degree of the meridian in various latitudes, calculated on the hypothesis that the ratio of the axes of the earth is that of 231 to 230. 293. The subject of attraction is discussed by Simpson in his work, entitled, The Doctrine and Application of Fluxions. I have not seen the first edition of this work, which appears to have been published in 1750. The second edition is dated 1776, which is subsequent to the author's death : 1 presume that this is a reprint of the first edition. This contains 576 octavo pages, besides the Title, Dedication, and Preface on xi pages in the first volume, and the Title of the second volume. Section IX. on pages 445... 479 is entitled, The Use of Fluxions in determining the Attraction of Bodies under different Forms. We have investigations, on the ordinary law, of the attractions of a straight line, of a circular lamina on an external particle which is perpendicularly over the centre, of a cone on a particle THOMAS SIMPSON. 187 at the vertex, of a cylinder on a particle on the axis, and of a sphere on an external particle. With respect to the circular lamina and the sphere, the investigation is also given for the case in which the attraction varies as the n ih power of the distance. The processes are all satisfactory, though some of them are rather artificial. The attraction of an oblatum on a particle at the surface is determined in essentially the same manner as in the Mathematical Dissertations ; but the analysis is a little simplified in some parts. In the Dissertations Simpson resolves the attraction in the direc- tions of the tangent and the normal ; in the Fluxions he resolves it parallel to the axes of the generating ellipse. Simpson remarks on his page 455 that the integral which we have considered in Art. 280 might be expressed in finite terms instead of an infinite series ; and this is obviously true. On his page 463 Simpson demonstrates exact results cor- responding to the approximate results enunciated by Stirling : see the diagram to Art. 153. Simpson shews that if PH be the direction of the attraction at P, then If divides C6r in a con- stant ratio, and the attraction varies as PH. These results may be established immediately by the aid of the modern formulae which are given in Art. 261. On his page 466 Simpson determines the attraction of an oblatum on any internal particle. This enables him to give a- more elaborate investigation than that in his Dissertations of Newton's postulate. On his page 474 Simpson gives 2 hours 26 minutes as the least time in which the Earth, supposed a homogeneous fluid, could rotate : this however might have been stated in the Dis- sertations, as the necessary elements for the result are there supplied. It corresponds to Laplace's '1009 of a day : see the Mecanique Celeste, Livre in. 20. The Table which we have given, from the Dissertations, in Art. 285 is not reproduced in the Fluxions. 294. Thus we see that the contributions of Thomas Simpson to our subject are of eminent importance. In the homogeneous 188 THOMAS SIMPSON'. Figure of the Earth lie first determined tin 1 existence of a limiting angular velocity, for which the relative equilibrium is possible; ami lu- implicit! v shewed that different ohlata mi'jht correspond to the same angular velocity. In Attraction lie gave an accurate investigation lor the cast- of an oblatum when tlie attracted par- ticle is at the surface; and also an approximate investigation for the case of any nearly spherical body of revolution, and the analysis which he employed would not have been unworthy of Laplace himself. Thomas Simpson was a mathematician of the highest order; and his merit is increased by reason of the great difficulties which impeded him. He has been pronounced "an analyst of first-rate genius," bv one who like himself had risen to distinction in spite of adverse circumstances, and whose life like his closed pivma- turelv in <_;'loom and trouble. He has been placed at the head of the non-academical body of English mathematicians by a member of that body, whose ability and learning well qualified him for form ng an opinion. It mav be doubted whether the eighteenth century, after the death of Newton, supplies any mathematician in England mure illustrious than the weaver whi.se genius raised him to the professorship of mathematics at Woolwich. See the life prefixed to Hutton's edition of Simpson's Select E.rf-rctsf-x ; Murphy's Theory of Equations, page -VI- ; Philosuphicul Magazine, September, 18-30, page 201). CHAPTER XL CLAIRAUT. 295. WE now arrive at the great work of Clairaut, which fs entitled Theorie de la Figure de la Terre, tMe des Principes de VHijdrostatique ; par Clairaut, de I A cademie royale des Sciences, et de la Societe royale de Londres, The work was published in 1743, and was reprinted in 1808. A note to the reprint states that the subject has been much con- sidered by mathematicians, and that the actual state of the theory will be found in the third book of the Mecanique Celeste; but on account of its historical interest the treatise of Clairaut may be studied with advantage, and so it has been reproduced without change or addition : the reprint in fact corresponds* nearly page for page with the original. It is stated that nothing has been neglected in order to remove the old errors of the press, and to avoid fresh errors : there is however an adequate supply of errors in the reprint. A reason for adding no notes is assigned in these words : "Elles auraient denature' un ouvrage original, sans le rendre plus utile au public." The principle involved in these words is known to have been held by Laplace ; and the conjecture has occurred to me that the reprint of Clairaut's work might have been suggested or encouraged by Laplace. The reprint is said to have been edited by Poisson: see the Catalogue des ouvrages.^de Simeon-Denis Poisson, 1851. I proceed to give an account of Clairaut's work ; I use the edition of 1808 : both editions are in octavo. The preliminary H>0 < LAIIUTT. nuto tn which I have just referred is of course peculiar to the edition d' hsov it occupies two pa^vs: a Dedication to tin- Coiute de .Maurepas occupies two pact's; tlicii an Introduction follows nn pages vii...xl : and the text, including a Table <>f ( 'hapters, occupies pai;VS 1 . . ..')(),S. l!. ( H>. The introduction gives a general notion of tlic subject ot 'the work. Let us brier! v consider what was the state of know- ledge iii 174o. With respect to fluid equilibrium Newton's prin- ciple of columns balancing at the centre, find llnvgens's principle of the plumb-line were allowed to be itecensuri/, but it was not known what principles were xiijfh'tdit, Maclanrin had advanced far in the theory of the attractions of ellipsoids of' revolution ; and had well discussed the homogeneous li^ure of the Karth: and from the fact that his researches appeared oriyinallv in Latin thev obtained a currency which the impoilant additions made to the theory by Thomas Simpson, published only in Kn^lish, probably never enjoyed. Ihe measurement of a decree r>f the meridian in Lapland had been made, and from a Comparison of this witli the measurements made in France, it had been inferred that the ratio of the axes of the earth was that of 177 to 17^; but the return of the expedition which had been sent to Peru was anxiously ex- pected, in order to obtain more information on this point : see Clairaut's pa^'es -'):), oO k The diminution of ^'ravitv in proceed- ing from the equator to the pole wa> well eslablisheil ; and it was ]lain that the whole diminution of gravity mu>t be greater than of the "'ravity ai the i>ol(j : see Clairaut's na (> 'e i! ( .'7. L'oO i!!)7. r lne C'artesians, according to ('lairaut. eidightenc'd by Xowton hdd that all bodies wore attracted to the centre <>f the Karth b\- a fnree which \ - aned inversely as the square tit the <bs- tance ; from this ( 'lairaul infers thai tin- ratio of (he axes of the Karih would be that of "*7<I 1" ">77: v '<' Clairaut's jia^es xiv, xvii, 1 l'>: in fact ( lairaut shews <<\\ \\\- pai_;'e 1 lo that this is true whatever Iti- fhe law of attraction prodded tlie direction always passes through the centre : si e also Art. ."(!. lint if we admit with Nev.'hiii that every particle ot" matter attract- every oth< r particle with a fbrce \~a IA'IIIL;' iiiversely as the CLAIRAUT. 191 square of the distance, bodies will no longer necessarily be attracted exactly towards the centre of the earth ; the direction of the resultant attraction on any particle will depend on the form of the earth, and on the position of the particle. Clairaut states the result which is demonstrated in the work, that considering the earth a homogeneous fluid in relative equilibrium the ratio of the axes will be that of 230 to 231 : see Clairaut's pages xxiii and 195. Clairaut remarks that the Newtonians may consistently with their fundamental principle obtain other results besides that just given ; for they have only to suppose that the earth is not homo- geneous. Clairaut considers that the result already given is that which the Cartesians ought to hold as following from their principles ; but he suggests for them various expedients by which they might escape from the conclusion : see Clairaut's pages xxiv, xxv. 298. Clairaut draws attention to his own methods for dis- cussing the equilibrium of fluids. He says Bouguer first re- marked that there are hypotheses as to the nature of attraction under which a fluid could not be in equilibrium : see Clairaut's page xxxi, and our Art. 219. Clairaut says on his page xxxiii : J'ai bientot reconnu qu'il etait vrai, ainsi que je 1'avais soupgonne", que 1'accord des deux principes ordinaires, c'est-a-dire 1'equilibre des coloimes et de la tendance perpendiculaire a la surface, n'assurait pas 1'equilibre d'une masse fluide ; car j'ai trouve qu'il y avait une infinite d'hypotheses de pesanteur ou ces deux principes donneraient la meme courbe, sans que pour cela les efforts de toutes les parties du fluide se contrebalangasseiit mutuellement. J'ai trouve ensuite deux methodes generales et sures, pour reconnaitre les hypotheses de pesanteur dans lesquelles les fluides peuvent etre en equilibre, et pour determiner la figure que les planetes doivent avoir dans ces hypotheses. The two general and sure methods to which Clairaut alludes in the preceding extract may be called the Principle of Canals t and the Principle of Level Surfaces : we shall give an account of them in our analysis of the work. It would appear from Clairaut's words on his page xxxiv, that he intended to furnish some ex- 192 CLAIRAUT. planation of these methods in his Introduction ; but the intention is not carried out, and the Introduction terminates somewhat abruptly. 299. The following points of interest in the Introduction may be noticed. On page xiii. Clairaut says in a note : Je fais ici la meme distinction que M. de Maupertnis (la Figure de la Terre determinee, etc.) entre la pesanteur et la gravite ; j'entends par pesanteur, la force naturelle avec laquelle tout corps tombe, et j'appelle gravite la force avec laquelle ce corps tomberait, si la rotation de la Terre n'alterait pas son effort et sa direction. I have already drawn attention to the distinction here ex- plained : see Art. 25. It must however be observed that Clairaut does not adhere strictly to the language which he here professes to adopt. Thus on his page 28 he uses pesanteur, and on his page 30 he uses gravite, meaning the same thing in both cases, namely my attraction; and on his page 144 he uses gramU where he ought to use pesanteur. On his page xxix. he enunciates the theorem which we call Clairaut' s Theorem : see Art. 171. On his page xxxviii. Clairaut is treating of rotation. He has supposed that an atom has described in an infinitesimal time a straight line Mm, so that if left to itself it would describe in the next equal infinitesimal time a straight line inn in the prolongation of Mm and equal to Mm. Then he says :...au lieu de la force qu'il aurait pour parcourir mn, on peut lui en substituer deux autres... Thus he uses the word force where we should now use velocity. In reading Clairaut's work, we are struck with the fact that although his conclusions are correct, his language is sometimes extremely inaccurate according to our modern notions. 300. Clairaut's work is divided into two parts. The first part treats of the general principles of fluid equilibrium ; the second part treats of the Figure of the Earth and the other planets, assuming the ordinary law of attraction. The first part consists of twelve Chapters, and occupies pages 1...151; the second part consists of five Chapters, and occupies pages 152... 304. f'LAIRAUT. 301. Clairaut's treatment of the theory of fluid equilibrium is a great advance beyond what his predecessors had given ; but it is not free from obscurity. Clairaut never uses, as we now do, a symbol p to denote the pressure at any point of the fluid ; this important step was first taken by Euler in the Berlin Memoires for 1755. I am little likely to undervalue any improvement in the Calculus of Variations, but I attach less importance to the well-known introduction of the symbol 8 into that subject by Lagrange, than to the introduction of the symbol p into Hydro- statics by Euler. . Before Euler thus illustrated the subject, there had been demonstrations in Hydrostatics, but I cannot consider that these demonstrations were altogether intelligible. 302. Clairaut's first Chapter occupies pages 1 . . .16 ; it expounds what may be called the Principle of Canals. Let there be a mass of fluid in equilibrium ; we may imagine any portion of it to become solid, and the remainder will still be in equilibrium. Thus we may solidify all the fluid except that contained in an. infinitesimal canal ; and so the fluid in such a canal will remain in equilibrium. This canal may be of any form, straight or curved ; it may pass completely through the mass, or it may be altogether within the mass returning to itself. The principle of canals had already in effect been used by Newton, Huygens, and Maclaurin ; though in general straight canals, which for distinction I call columns, had sufficed for their purposes ; see Arts. 24, 55, and 245. Although the Principle of Canals as stated in the preceding Article will be admitted to be obvious, yet Clairaut's method in applying the principle is not always clear. Thus, for example, on his page 2, he has a canal ORS passing entirely through a mass of fluid, which is in equilibrium ; he says : "or cela ne peut arriver que les efforts de OR pour sortir vers S, ne soient e*gaux a ceux de SR pour sortir vers 0." But how are we to measure the efforts which OR makes to escape towards S; or in fact what distinct idea can we form of these efforts ? Again take an example from his page 12. He has two canals of fluid HI and KL under certain circumstances ; and he says that T. M. A. 13 194 CLAIRAUT. the weights of these two canals will be the same. But it is not immediately obvious how these weights are to be measured : the fact in modern language is that the pressure at H is equal to the pressure at K, and the pressure at / is equal to the pressure at L. 303. Clairaut's second Chapter occupies pages 16... 28; it consists of general reasoning to shew that under certain attractive forces a fluid mass will remain in equilibrium. The Chapter seems superfluous, for in the sixth Chapter we have substantially a more satisfactory treatment of the subject. In reading the second Chapter it may assist the understanding if we conceive the fluid to be all enclosed within a rigid envelope; and then the sixth Chapter will in fact shew that we may dispense with this envelope. 304. Clairaut's third Chapter occupies pages 28... 33; it considers a law of attraction under which a fluid mass could not be in equilibrium. The law is that in which the attraction towards a fixed centre is not a function of the length of the radius vector alone, but also of the position of the radius vector. The following is the demonstration, translated into modern language, of the impossibility of fluid equilibrium under such a law of force. Let MN\)e an arc of a circle having the centre of force 6 Y for centre; let P Q be an arc of a concentric circle, such that MFC is a straight line, and also NQC a straight line. Conceive the fluid in an infinitesimal canal MN to become solid ; take the moments round C of the forces which act on it: thus we see that for equilibrium the pressure at M must be equal to the pressure at N. Similarly the pressure at P must be equal to the pressure at Q. But since the attraction along PM is not the same at equal distances from G as the attraction along QN, the change of pressure in passing from P to M is not equal to the change of pressure in passing from Q to N. This contradicts the former result. 305. Clairaut infers that there are innumerable cases in which a fluid mass will not be in equilibrium even although the con- ditions of Newton and Huygens are both satisfied. Clairaut is brief; we may expand his remarks. Let there be a curve r = <f)(6) which revolves round the initial line ; suppose we want to have a CLAIRAUT. 195 mass of fluid in relative equilibrium when rotating with a given angular velocity round the initial line under an attractive force to the pole, and taking the form of the solid of revolution just generated. Since the angular velocity is given, the centrifugal force is known at every point of the boundary ; hence the amount of the attractive force can be determined which must act at any point of the boundary, along the radius vector, so as to satisfy Huygens's principle of the plumb line : let ty (6) denote the amount of this attractive force at the point for which 6 is the angular coordinate. Assume for the formula of attractive force f(0) {<f> (&) - r} n + ^ (6), a function of r and 0, in which f(0) is at present undetermined ; then it is obvious that Huygens's principle is satisfied. To satisfy Newton's principle we require that the /*(*) expression I [/($) (< (0) r } n + ^ (#)] dr, which measures the / o weight of a column, should be constant, the integration being r i (f)\\n taken with respect to r. This gives J ^ V) W\ '> equal to a constant; and so/(0) is determined. Thus Newton's principle is also satisfied. But by Art. 304 the fluid cannot be in equilibrium under the law of force which we have assigned. 306. Clairaut's fourth Chapter occupies pages 33... 39; it determines the form of a mass of fluid in relative equilibrium acted on by certain forces. Suppose fluid to rotate round the axis of x, with angular velocity co, under forces of which the acceleration parallel to the axis of x is X y that parallel to the axis of y is , and that parallel to the axis of z is - ; where r 2 = y^ + z* : then the equation to the free surface when there is relative equilibrium is (Xdx + Edr) -f ^ = constant, 2* and the condition -f- = -7 must hold. dr ax This is not quite Clairaut's notation, but the difference is unimportant. 132 196 CLAIRAUT. The demonstration of these results will be found in our ordinary treatises on Hydrostatics. I do not regard Clairaut's process as quite satisfactory until it is translated into our modern language. Clairaut, after giving the equation of condition which we express as =- = -y- , says briefly and authoritatively : " Toutes les fois que cette Equation aura lieu, on sera sur qu'il y aura equilibre dans le fluide." To me there appears some difficulty at this point in the theory of the equilibrium of fluids. We can shew clearly that certain conditions must hold for equilibrium ; but it is not quite obvious that if these conditions are satisfied there will be equili- brium. Our modern writers seem to shrink from making the posi- tive assertion of Clairaut, though perhaps sometimes it is implicitly adopted. But it is obvious that Clairaut asserts too much. Sup- pose for simplicity we restrict ourselves to one plane, and put X and Y as usual for the forces : it is not sufficient for equilibrium that -,- = -y . For example take X= ^ ., , and Y= ; dy dx x+y +y let p denote the pressure, and p the density as usual. Then we ydx xrfy , , .y get a = p - 5 r 2 \ ana therefore = p tan - + constant. x + y* x But this value of p is not admissible, for it would involve discon- tinuity, that is more than one value of p at the same point. See D'Alembert's Opuscules Mathematiques, Vol. v. page 10. In fact Clairaut's own pages 83... 90 are sufficient to shew that his lan- guage is too positive. 307. The condition -, = -r- ensures that Xdx -f Edr is a complete differential. The notion of a complete differential, and the appropriate condition, seem to have been first introduced by Clairaut himself: he refers to his memoir on the Integral Calculus in the Paris Memoires for 1740. Clairaut explains thus, in a note on his page 38, one of the symbols which he uses : " On entend par -=- la differentielle de la fonction P, prise en supposant x seulement variable, et dont on a CLAIRAUT. 197 6te les dx" It seems more natural to take the differential coefficient as the prior and simpler conception, and not the differential, as Clairaut here does. 308. Clairaut's fifth Chapter occupies pages 40... 52; it introduces the use of Level Surfaces ; these were first considered by Maclaurin ; see Art. 248. Clairaut calls a level surface a surface courbe de niveau; and the space comprised between two level surfaces he calls a couche de niveau. Clairaut gives the following proposition : suppose a mass of fluid divided into an infinite number of infinitesimal shells ; if at any point of every shell the thickness of the shell is inversely proportional to the resultant accelerating force, the fluid will be in equilibrium. I cannot say that Clairaut's reasoning satisfies me. Indeed even with the modern methods, although it is easy to shew that when fluid is in equilibrium the thickness of the infinitesimal shells must follow the law assigned, yet to shew decisively that when this law of thickness holds the fluid must be in equilibrium seems far from easy : see Art. 306. Some remarks on Clairaut's reasoning will be found in the Cambridge Mathematical Journal, Vol. II. pages 18... 22. However, granting the proposition, Clairaut very ingeniously deduces the same equation as before for the free surface of a mass of fluid in relative equilibrium ; and also the same condition as before connecting the forces : see Art. 306. Another example of the strange mode of expression which we find in the book occurs on Clairaut's page 51. If we take an infinitesimal canal within an infinitesimal level shell we say in modern language that the pressure is constant throughout the canal ; Clairaut speaks of la liqueur. . ., ne pesant point. 309. Clairaut's sixth Chapter occupies pages 52... 63; it supplies examples in which the equation to the free surface of fluid in relative equilibrium is found when given forces act. In. one example fluid is supposed to rotate round a vertical axis, the velocity of rotation being a function of the distance from the axis. Clairaut refers to two solutions which had already been proposed for this problem ; namely, a correct solution by Daniel Bernoulli 198 CLAIRAUT. on pages 244, 245 of his Hydrodynamica, and an incorrect solu- tion by Hermann on page 372 of his Phoronomia : see Arts. 98 and 230. In another example Clairaut supposes the fluid to be attracted to any number of fixed centres. In another example Clairaut supposes the particles of fluid to attract each other with a force varying as the distance ; and the fluid to rotate round an axis : in this case the free surface is that of an oblatum. Clairaut uses the known theorem that under such a law of attraction the resultant attraction varies as the distance from the centre of gravity of the whole attracting body : see Art. 12. 310. Clairaut's seventh Chapter occupies pages 63... 77; it discusses a problem in fluid equilibrium proposed by Bouguer. We will give an account of the substance of the problem by the modern method. Let x, y, z be the coordinates of any point of a fluid ; let the shortest straight line be drawn from this point to a given surface ; let r be the length of this straight line, and x, y, z the coordi- nates of the point where it meets the given surface. Let the force acting on the fluid at (x, y, z) be along the line of r, and be denoted by/ It is required to determine the pressure at any point, and the form of the free surface. In modern notation we have \ dp _x x * 1 dp _ y y - 1 dp "~ J* ' ~~ _ z z r . r ~~ Now rdr = (x - x) (dx r - dx} + (y' - y) (dy' - dy) + (z' - z} (dz' - dz} ; and (x - x} dx' + (y' - y) dy' + (z r - z} dz' = 0, because r is the shortest distance between (x, y, z) and the given surface; hence rdr = (x x) dx(y y) dy - (z z)dz; therefore - dp = -fdr. GXAIRAUT. 199 Hence /must be a constant, or a function of r ; say/= </> (r), and - = - ^r(r) 4- constant, where ^ (r) is the integral of </> (r). Thus the pressure at any point of the fluid mass is determined, and the form of the free surface is found by making the pressure constant. Of course this is not Clairaut's method, as we have already remarked that he does not use a symbol for the pressure. He restricts himself to the case in which the given surface is a surface of revolution. Clairaut considers that in order to render the hypothesis natural we must suppose there to be a central solid mass; for otherwise we should have some particles of fluid indefinitely close to each other, and yet acted on by forces the directions of which include a finite angle, ce quiest choqiiant. 311. Clairaut gives a second solution of the problem by a kind of general reasoning ; see his page 69. He restricts himself to the case in which the given surface is a surface of revolution ; and so, instead of considering normals to a surface as we did in the preceding Article, he considers normals to a given curve. Take a second curve, the points of which have a constant shortest distance from the given curve ; that is, take a second curve which has the same evolute as the given curve : then it follows from the preceding Article that the pressure is constant throughout the second curve. Clairaut arrives, in his own way, at a result which corresponds to this; he expresses it, however, by saying that le poids de OT doit etre nul; where OT denotes an infinite- simal canal in the form of our second curve. 312. Clairaut's eighth Chapter occupies pages 78... 93; in modern language, we may say, that it is a modification of the sixth Chapter, by using polar coordinates instead of rectangular : thus confining ourselves to one plane, instead of Xdx + Ydy we now get Mr + TrdO. The most interesting part of the Chapter is what Clairaut 200 OLAIRAUT. calls the explanation of a species of paradox. The general equa- tion to the free surface of the fluid is I Edr + I TrdO = constant ; the paradox consists in this, that Newton's principle of balancing rr columns gives / Rdr = constant for the equation to the free sur- J Q face, which may in some cases differ from the former result. We will omit all reference to the rotation of the fluid. Sup- pose, for example, that R = r6*, and TrQ\ then the two re- suits agree: so also they agree if E - , and T= - r 6 But suppose that R= 2+ ^ , and T= ^ 2 + ^ ; then ac- cording to Newton's principle of balancing columns we get V(r 2 +0 2 )-0 = constant; while the other result is ^(r^+B 2 ) = constant. Clairaut's explanation consists of reasoning to shew that the latter result is correct ; but it does not appear to me that he is happy in his explanation. Such a force as .-r-, 7^ i g incon- ceivable when r = ; and thus to render his problem reasonable, a portion of the fluid round the -origin must be supposed to become solid ; and then Newton's principle of columns balancing at the centre is no longer applicable. D'Alembert objects, with justice, to Clairaut's explanation: seethe Opuscules Afathematiques, VoL v. pages 11 and 15. Similar remarks to those in Art. 306 are applicable here. It is not sufficient for equilibrium that Rdr + TrdO should be a perfect differential. Suppose, for instance, that this is the differ- ential of a function f(r, 6) ; then if, when r 0, the value of f ( r > ^0 still involves 0, the pressure is not the same in all direc- tions round the origin. Not one of Clairaut's three examples could correspond to the equilibrium of a free surface. Suppose, for instance, that T=rd\ then when 6 increases by 2?r, we get a different value of T at the same point. But there might be equilibrium in a portion of the fluid confined, when necessary, by fixed planes. CLAIRAUT. 201 313. Clairaut's ninth Chapter occupies pages 94... 105 ; in this Chapter the results are extended to space of three dimen- sions, which in the previous Chapters had practically been applied only to space of two dimensions. Thus with the modern usual notation Clairaut finds that the free surface of the fluid in equili- brium must be such as to make the integral of Xdx 4- Ydy + Zdz a constant ; and, moreover, the following conditions must hold : dX = dY dX = dZ dY = dZ dy dx ' dz dx* dz dy ' These conditions are satisfied for such forces as occur in nature ; so that Clairaut arrives substantially at this result: a mass of homogeneous fluid, under the influence of such forces as occur in nature, will be in equilibrium if Huygens's principle of the plumb- line holds at the free surface. 314. Clairaut's tenth Chapter occupies pages 105... 128 ; it is on capillary attraction. Clairaut gives only extreme generalities. He may be said to shew that it is not impossible, and even not improbable, that the phenomena may be explained by supposing particles of fluid and particles of a solid tube to attract an ad- jacent particle of fluid with forces which are sensible only at a very small distance. But the Chapter is too remote from my subject to warrant me in examining it closely. Laplace devotes a paragraph to Clairaut's theory of capillary attraction in the Mecanique Celeste, Livre XI. 1 ; Laplace's opinion is not favour- able, he says: "cette the'orie me parait insignifiante...." 315. Clairaut's eleventh Chapter occupies pages 128... 138; it treats of the equilibrium of fluid which is not homogeneous. In modern language, Clairaut undertakes to shew that level sur- faces must be surfaces of equal density : we now know that this proposition is not necessarily true, unless Xdx + Ydy + Zdz is a perfect differential. To this D'Alembert seems to refer in his TraiU...des Fluides, second edition, page 50. When a mass of fluid, like a planet, is not homogeneous, but yet is in equilibrium, Clairaut considers that the denser shells must be below the rarer ; see his pages 134, 138, 280, 292. He 202 CLAIRAUT. does not demonstrate this condition, which is theoretically not necessary for equilibrium, though it may be essential for stable equilibrium. 316. Clairaut's twelfth Chapter occupies pages 139. ..151 ; it shews how we may determine the law of attraction at the surface of the Earth, from the results given by observation. By pendulum experiments we determine the force of gravity at any point on the Earth's surface ; by measuring various lengths of degrees of the meridian we ascertain the form of the Earth's surface, and thus we can deduce the effect of the centrifugal force at any point : then knowing the values of gravity and of centrifugal force at any point, we can obtain the attraction at that point. But this does not determine the law of attraction within the surface of the Earth ; so that on this point we must endeavour to make some natural hypothesis by the aid of the theory of fluid equilibrium. Assuming that the Earth is a homogeneous fluid, and that the direction of attraction always passes through the centre, Clairaut gives a simple proof that the ratio of the axes must be very approximately that of 576 to 577, whatever be the law of attrac- tion; see Art. 56. Hence, assuming that the ratio of the axes as determined by the French and Lapland arcs is really that of 177 to 178, it follows that the direction of attraction cannot always pass through the centre. As an example Clairaut takes the ratio of the axes of the Earth to be that of 177 to 178 ; and he assumes that the dimi- nution of gravity in passing from the pole to the equator varies as the square of the cosine of the latitude, the total diminution being of the polar gravity : these facts depend on observations in France and Lapland. Then he shews that these data are con- sistent with an hypothesis of the law of force belonging to Bouguer's class: see Art. 310. This example is worked out in detail by Clairaut ; but though not destitute of interest theoreti- cally, it is of no practical value. 317. We now arrive at Clairaut's second part, which is that with which we are specially concerned. It consists of some intro- CLAIRAUT. 203 ductory observations, followed by five Chapters. The introductory observations occupy pages 152... 158. Clairaut refers to his own former memoirs in the Philosophical Transactions, which we have noticed in our Chapter VI. Clairaut's researches on the figure of the Earth, considered homogeneous, arose from his desire to demonstrate Newton's postulate: see Art. 44. Clairaut's researches on the figure of the Earth, con- sidered heterogeneous, arose from his desire to test and correct a remark made by Newton, namely, that the Earth if denser towards the centre would be more flattened than if it were homo- geneous : see Art. 30. Although the case of the homogeneous figure of the Earth could be deduced by a single substitution from the formulae given by Clairaut for the heterogeneous figure, yet he judged it con- venient to treat separately the homogeneous figure ; and for this purpose to abandon his own method and follow that given in Maclaurin's Fluxions. 318. Clairaut's first Chapter occupies pages 158. ..198; it contains the theory of the homogeneous figure of the Earth or a planet. This is essentially the same theory as Maclaurin gave ; but it is more easy to follow by being broken up into short sec- tions, and printed in a more pleasing manner. The exact values of the components of the attraction of an oblatum on a particle at its surface are given ; the components being estimated parallel and perpendicular to the axis of revo- lution. Clairaut holds that a rotating mass of fluid in relative equi- librium must assume the form of an oblatum; see his page 171. We have already observed that Maclaurin and Thomas Simpson in like manner asserted more than they were able to demonstrate: see Articles 249 and 283. On his pages 188... 190 Clairaut shews that the gravity varies as P6r, to use our notation in Art. 153 ; but instead of the simple method which we adopt there, Clairaut first demonstrates the proposition of Art. 33, and then deduces the required result. 204 CLAIRAUT. The relation which connects the ellipticity of the Earth with the value of the ratio of the centrifugal force to the attraction can be expressed exactly, or approximately in various forms, according to the notation adopted : see Arts. 262 and 283. The following is the approximate result in Clairaut's notation : he takes the ratio of the equatorial axis to the polar axis to be that of 1 + 8 to 1 ; and he uses <> to express the ratio of the centrifugal force at the equator to the gravity there, not to the attraction: then 4 2 8 from which 1 1 His S is our --^T- -- 1 ; and his (f> is our J . . He finds for the value of $ ; see his page 194, from .... , 1000 which he gets 8 = 319. On his pages 195, ..198, Clairaut applies his formula to determine the ellipticity of Jupiter ; he arrives at the conclusion that the ratio of the axes is that of 100J to 90J. This differs very little from Newton's final value : see Art. 29. Modern observation gives a much smaller value to Jupiter's ellipticity than that which Newton and Clairaut derived from theory. Sir J. Herschel in his Outlines of Astronomy, 1849, Art. 512, states the ratio of- the axes as that of 107 to 100 ; he adds : And to confirm, in the strongest manner, the truth of those princi- ples on which our former conclusions have been founded, and fully to authorize their extension to this remote system, it appears, on calcula- tion, that this is really the degree of oblateness which corresponds, on those principles, to the dimensions of Jupiter, and to the time of his rotation. In the edition of 1869 the ratio is changed to that of 106 to 100 ; but the passage just quoted remains unchanged. It is CLAIRAUT. 205 obvious that the remark cannot be accepted. For in the first place, if we consider Jupiter to be homogeneous, theory and observation are by no means in correspondence ; secondly, if we suppose Jupiter not to be homogeneous, we shall be compelled to make some arbitrary hypothesis respecting the internal constitution of the planet, and cannot therefore appeal to the result as con- firming in the strongest manner the truth of our principles ; and 7 thirdly, if a calculation once gave - as the ratio of the difference of the axes to the minor axis, we cannot afterwards assert that the f* calculation gives - as the ratio. 320. Clairaut's second Chapter occupies pages 198... 232; it treats of the relative equilibrium of rotating homogeneous fluid which surrounds a spheroid composed of strata of varying density. We have first a theorem respecting the attraction of a circular lamina on an external particle which is so situated that its projection on the lamina is very near the centre. Take the centre of the circle as the origin; let the axis of x pass through the projection of the attracted particle, and let h denote the distance of this projection from the centre, and k the distance of the particle from its projection; let r denote the radius of the circle, T the thickness of the lamina, and p the density. Then the attraction resolved parallel to the axis of x, estimated towards the origin, the integration being taken over the area of the circle. Integrate first with respect to x ; the limits may be denoted by - % and f : thus we get The process is exact up to this point. If we suppose h very small, we may expand the expression under the integral sign in 200 CLAIKAUT. powers of 7^; and thus we get :lprJt | - , that is <7y. But 2 (/// is equal to the area of the circle; (/* + // J - ^ ! thus we obtain finally p~Ji r ., . , ------- x the area of the circle. Tin: 1 investigation would apply to a lamina which is nearly though not exactly circular, and leads to the same result. Clairaut's own process is given in a geometrical form, but it is substantially equivalent to ours. Wo proceed to make use of the result. 321. Clairaut requires the approximate value of the attraction of a nearly spherical ohlutum on an external particle. Let C denote the centre of the oblatum, and J/ the external particle, The attraction may be resolved into components along M(\ and at ri;.di! angles to ML 1 , it is sufficient for Clairaut's purpose to consider the attraction along jj(' to be the same as if the mass of the oblatum were collected at C. To find the attraction at right angles to MC, he calculates the aggregate effect by the aid of the result in Art. 320. Let the diagram represent the ellipse which is obtained by a meridian section of the oblatmn passing through .17. Through any point //in CM draw a chord at right angles to CM; the middle points of all such chords will be on a diameter. Let CA be the direction of this diameter, so thai I/A is the // of Art. '^<>, the chord itself being the inteiv.ection of a lamina, at right angles to L'Jf with the meridian plane. Lot CJI = x, and the angle JK'K = /3, so that // = ,/ tan /?. Let f/A r } and CM-- r /. Now It is very small, because tan/:? is verv small; and thus, without introducing any error to the order of accuracy whi'-h \ye ad<>]>], we can use certain approximate values of the /'" i /c ;: . and the area, which occur in Art. IliM). We take 77 'V jj'j for i he n.t'cu, and 1,7 .r/' -{ c.~ j;* for t he /"' 4- k~. These CLAIRAUT. 207 approximations amount to neglecting the ellipticity of the oblatum ; and as we have the common factor tan /?, our error is of the order of the product of tan /3 into the ellipticity. Hence by Art. 320 we have for the attraction of the whole oblatum at M in the direction at right angles to MG and towards CK, IT, tan/3 [* *(*-*)** ,. H ( 7 -* 2 -fc 2 -a 2 * The value of the definite integral which occurs here is 5 7 4 ' Clairaut himself obtains it by a peculiar artifice. By modern methods we may proceed thus : Put 7 a +c 2 -27# = 2 2 , and let a = 7 - c, and b = y + c; then we find that J Integrate by parts, observing that 208 CLAIRAUT. thus we find that the integral ~v Thus the required attraction is -5 . The angle ft is exactly the same as the angle between the diameter CK and the n'ormal at its extremity, and is therefore very approximately equal to the angle between CN and the normal at N. 322. Clairaut introduces and defines the term ellipticity of a spheroid on his page 209 : with him it denotes the ratio of the difference of the equatorial and polar diameters to the polar diameter : so that taking 2a for the equatorial diameter and 26 for the polar diameter the ellipticity is t- . To the order however which is sufficient for our subject we might also define the ellip- ticity as - , and this is the sense in which we prefer to use tv the term. 323. We can now give an outline of Clairaut's investigations ; we shall however change his notation for a more modern one. Suppose the central part of the Earth solid, consisting of strata nearly spherical ; and outside this let there be homogeneous fluid. Let r denote strictly the polar semiaxis of a stratum, but with sufficient approximation in many cases the radius drawn from the centre in any direction to the stratum. Let p denote the density and e the ellipticity of this stratum ; let e' denote the ellipticity CLAIRAUT. 209 of the stratum which forms the boundary of the solid part. Let p l denote the density of the fluid, and v the ellipticity of the sur- face of the fluid. Let r be the value of r at the boundary of the solid part, and r t the value of r at the surface of the fluid. Let < t be the angle at the point corresponding to r v between the normal to the stratum and r r Thus the subscript 1 always indicates a value relative to the surface of the fluid. Since the fluid is homogeneous Huygens's principle furnishes us with the necessary and sufficient condition for equilibrium. At any point of the surface of the fluid we have a central force which we will call F t and a force in the meridian plane at right angles to the radius vector towards the equator which we will call T r ; there is also the centrifugal force which at the equator would be jFm our usual notation, and which will be jFsm\ at the place considered, if X denote the angle between the radius vector and the polar semiaxis. Hence resolving all the forces along the tan- gent to the meridian we have as the condition of equilibrium Famfa-T-jFswXcoaX^O ............... (1). We must now develope this equation. With regard to F it will be sufficient to consider the whole mass as made up of spheri- cal strata of varying density ; and thus 4/7r f ri *J z I pr dr. ?* Jo Next consider T. If there were a homogeneous oblatum of A O6 density p this force, by Art. 321, would be * 4 , where r denotes the radius vector of the oblatum in the direction of r,. For such an oblatum in which the radius vector is r -f- dr the force would be - 4 1 tan /3r s -f -7- (tan /3r*)drl . Hence the force arising i \ / from a shell of which p is the density and dr the thickness in the direction of r, is -^~, -7- (tan/3r s )dr. 5r t dr Thus we obtain T. M. A. 14 210 CLAIRAUT. where p, now supposed variable, indicates the density of the stra- tum corresponding to r. This method of treating strata of varying density occurs very frequently in our subject and should be carefully noticed. Now by the nature of an ellipse it follows that to the order of approximation which we here retain tan jB or sin ft is equal to 2esin\ cosX; and to that order ^ has the same meaning as /8,. Hence by simplifying we get from (I) The form of (2) may be modified by separating the integral into two parts, one extending from to /, and the other from r to r^ ; in the second interval the density is constant and is de- noted by p r Thus P = P 3- + ftVl - er. If we employ the second of these modifications, (2) becomes f r/ ^ d If we put A for I prVr, and D for I p -j- (er 5 ) dr we get, by employing also the first modification, ^ 2 (6Z>- 6p l y 5 ) +j (ISA + 5/y, 8 - 5p/ 3 ) 6 = 3 This is a very important formula in our subject; it agrees with that given by Clairaut on his page 217, allowing for a mis- print with him: the investigation is substantially like his though in form rather different. 324. The general result (3) of the preceding Article is applied by Clairaut on his pages 218... 222 to some special cases. CLAIRAUT. 211 I. Suppose the whole mass homogeneous ; then and we obtain e, = -? : . 4 this as far as it goes agrees with Art. 318. II. Suppose the solid part homogeneous as well as the fluid part, but the densities of the two parts different. Let the density of the solid part be denoted by p,(l -f K) ; then and we obtain 6 = 1 10/er' 8 + 4*r* We shall find hereafter that this result reappears in the Mfaanique Celeste, Vol. v. page 30. Clairaut remarks that if we consider e t to be known by obser- vation, this formula will guide us in making suitable hypotheses as to the radius, the elliptic! ty, and the density of the assumed solid central part. He warns us that if we suppose K to be negative we must remember that it is to be numerically less than unity ; but the result shews us that this is an inadequate restric- tion : for if K be negative it must not be numerically nearly equal 4r 3 to T7TT3 , and this might be much less than unity if r were nearly equal to r x . The truth is that if K be positive the above result may be accepted without scruple ; but if K be negative we must carefully examine whether the value of e t obtained from the formula is a small quantity. If in the above formula for e x we suppose e' = 0, and K negative, AT 3 and put 3- = - X, the result agrees with that obtained on page 156. r \ III. Suppose as a particular case of II, that the solid part is to be similar to the whole mass, and that we require the ellipticity 142 212 CLAIRAUT. to be greater than it is when the whole mass is homogeneous. Then put e, = e = - (1 +^>), where p is some positive quantity ; thus we deduce pr 3 , 8 3r 5 \' 2,7) so that /c is necessarily negative. IV. In the preceding result, suppose that the difference between r and r t is infinitesimal ; put / = r l (1 X), where X is infinitesimal : then K= ~ so that K differs only infinitesimally from unity. Thus we have the case of a film of fluid which surrounds a solid body of infini- tesimal density; the outer and inner surfaces of the film are similar, similarly situated, and concentric oblata. V. Instead of being a film as in IY. let us suppose the planet to be a shell of finite thickness ; and let the internal part, though hard, be supposed of no density or of no attracting power : then we must solve the equation 3 / , 8 r' 5 \ /5 , 3 r' 5 and .take for a positive value less than unity, if such a value r i should occur among the roots of the equation. VI. Now return to the suppositions in II. If the density of the central part is to be greater than the density of the fluid, O'} V & and 6 X to be greater than -j , then e' must be greater than - 1 . For put e' = (e l + 7) -^ , and substitute in the result given in IT. ; thus we get 5; 3y K r 3 ~ 4 2fcr' s + 2^ ' and the second term will not be positive unless 7 is positive. CLAIRAUT. 213 325. Clairaut applies the last result of the preceding Article to two criticisms on Newton. In the case of the Earth, if we wish to have e t greater than - , it is not sufficient merely to suppose a solid nucleus of greater density than the fluid ; it is necessary to have the ellipticity of this solid nucleus greater than -^ : see Art. 37. In the case of Jupiter, if we wish to have e t less than - , it is not necessary to suppose that the equatorial parts have been scorched by the Sun into a greater density than the other parts ; it is sufficient to suppose that the solid nucleus is denser than the y c: fluid, and that it has an ellipticity less than -~ l : see Art. 31. 326. Clairaut shews in his pages 224 and 225, that an oblongum may be a form of relative equilibrium. For in case II. of Art. 324, if e' is negative and numerically greater than , 5 ' * , then l is negative. But even if e is positive, it will be possible to have e t negative if K be negative ; Clairaut does not make this remark, to which D'Alembert seems to attach great importance ; see the Opuscules Mathtfrnatiques, Vol. VI. page 77. The fact simply is that Clairaut's general formula contains somewhat more than he himself verbally drew from it. 327. Suppose the depth of the sea to be not greater than the height of the mountains; then Clairaut considers that we may without sensible error regard the earth as an oblatum covered with a film of water; see his page 225. In this case he takes x = e, and r x = r ; and so the equation (3) "of Art. 323 becomes 214 CLAIKAUT. 328. It has been objected that Clairaut ought not to have supposed 6j=e': see D'Alembert's Opusc ules Mathtfmatiques, Vol. VI. page 75, and Cousin's Astronomie Physique, page 164. If then we put r x = r, but do not put e t = e', the equation (3) of Art. 323 becomes (WA - 2 ft O 6, = 1 (D - fte'O + 6AJ. As before then we may say that Clairaut's general formula contains more than he was contented to draw from it. But we must observe that if we suppose the stratum of fluid to be very thin, but do not take e, = e', the fluid will not necessarily cover all the solid : either the polar parts or the equatorial parts may be left without fluid. 329. Clairaut applies his result which we have given in Art. 327 to shew that if 6 = 6^-^ wj, where u is positive, and the density diminishes continually from the centre, the ellipticity will be less than when the mass is homogeneous. For, using this expression for e, we have HT^ ( ^^$^ = 36^-6, (pr^ + e.j^u^dr = 36^14-6^ say, where G is some positive quantity, since by supposition ~ is CUT negative. Thus we obtain, , - QA l + 4 *x & = 54;, so that which is less than -4-. 4 -_/ CLAIRAUT. 215 Clairaut expresses his result very awkwardly in words ; he says that the spheroid will be less flattened than in the homogeneous case, unless the ellipticity of the strata diminishes from the centre to the circumference, and in a greater ratio than the squares of the distances. The language would imply that the squares of the distances diminish from the centre to the circumference. He should have said, provided the product of the ellipticity into the square of the distance is never greater than at the surface. Clairaut on his pages 228.., 232 gives some special cases of the general result in Art. 328, by assuming special laws of density ; his results are accurate, and he points out the objection to some corresponding investigations of Maclaurin : see Art. 267. 330. Clairaut's third Chapter occupies pages 233... 262; it discusses the law of the variation of gravity at the surface of a spheroid of revolution composed of strata of varying density and ellipticity. Clairaut shews that the diminution of gravity in passing from the pole to the equator varies as the square of the cosine of the latitude ; and he establishes the theorem which we now call Clairaut's Theorem. We will proceed to give some details. 331. Suppose a particle placed outside a circular lamina; when the projection of the particle on the lamina is very near the centre of the circle, the resultant attraction on the particle is very nearly the same as if the particle were at the same distance from the centre of the circle, but had its projection coincident with the centre : Clairaut shews this briefly by general reasoning. If we proceed analytically as in Art. 320, we shall find that when the particle is displaced so that its projection moves from the centre to a distance h from the centre, the attraction perpendicular to the lamina is not changed to the order h, while there is a transverse attraction produced of the order h ; so that the change in the resultant attraction is of the order h*. The result holds also for an ellipse or any other central curve. 332. If a circular lamina, and an oval lamina which is nearly circular, have the same centre and the same plane and equal areas, they exert approximately the same attraction on a particle, the 216 CLAIRAUT. projection of which would coincide with the common centre : Clairaut shews this briefly by general reasoning. 333. The propositions of the two preceding Articles lead Clairaut to the following general result. Let C be the centre of an ellipsoid of revolution nearly spherical, and M an external particle ; let MG cut the solid at N t and let MG produced cut the solid at n ; the attraction of the solid on a particle at M is approximately the same as that of an -ellipsoid of revolution of equal volume having Nn for its axis of revolution. The original solid may be an oblatum or an oblongum ; which- ever it be the derived solid will be sometimes an oblatum and sometimes an oblongum, according to the position of the straight line CM. It must be observed that the approximation holds as far as the first power of the ellipticity inclusive; in fact the errors in Arts. 331 and 332 are of the order of the square of the ellipticity. 334. Clairaut then finds the attraction of an ellipsoid of jevolution which is nearly spherical on a particle which is on the prolongation of the axis of revolution. I have already adverted to the method which he uses : see Art. 165. The result is correct to the first power of the ellipticity inclusive. 335. The pages 233... 243 of Clairaut's work which we have just considered were, in substance, originally published in the Philosophical Transactions ; see Art. 164 : these pages well deserve perusal as a good specimen of the ingenuity and simplicity of Clairaut's investigations. A modern student will probably like to verify by analysis the important result in Art. 333. The simplest way perhaps is to find the potential of the original ellipsoid of revolution on the particle at M t and shew that it is equal to the potential of the derived ellipsoid of revolution, so far as terms of the first order inclusive. Take C for the origin, CN for the axis of z ; let the axis of y be the diameter which is, conjugate to Nn in the meridian plane of CLAIllAUT. 217 the given ellipsoid which contains CM ; let the axis of x be at right angles to those of y and z. Let C3L= 7 ; let ^ denote the angle between the axes of y and 2. Then the potential of the given ellipsoid is the limits of the integration are determined by the equation to the given ellipsoid, which we may take as x 2 f z* -i + tf + -* = ! a 6 c* Put r 2 for 3? + ?/ 2 + (z 7) 2 ; then we may expand the term under the integral signs in the form 1 y(z-j) cos x , 3 if (* - y)* CM* % r r s . "^2 " r 5 The second of these terms gives zero as the result when inte- grated, because y is as often negative as positive. Thus if we reject the squares and the higher powers of the small quantity cos x> the potential becomes dx dy dz Assume x a^, y=bq, = cf; then the potential can be transformed to where F stands for ale sin ^, and the limits of integration are determined by ? + ?+?-!. Now when we form the potential of the derived ellipsoid we obtain, if h denote the two equal semiaxes, 218, CLAIRAUT. the limits being the same as before. And by the condition of equal volumes we have dbc sin ^ = h 2 c ............ v ................ (3). Since the original ellipsoid is nearly spherical we have a = c(l+X) and = c (!+/*) where A. and //, are small, being of the order of the ellipticitiesi Thus from (3) we have tf = c 2 (1 + X + IM) sin x, but sin % being the sine of an angle, nearly a right angle, we shall find that it differs from unity by a quantity which is of the order of the squares of the ellipticities. Thus to our order and so we have to our order _ Hence since a 2 = ^ + ^ > and " = we see that to our order (1) becomes Expand the denominator under the integral sign in powers of o (f 2 ~~ if) > then the term under the integral sign which 2> involves the first power of this small quantity obviously vanishes by symmetry : so if we neglect the square of a 2 & 2 , the expres- sion (1) reduces to the form (2). This is the required result. 336. "We are now prepared to find the value of gravity at any point of the surface of our hypothetical Earth. Suppose r the polar radius, r (1 + e) the equatorial radius of an oblatum, where e is small; let p be the density. We shall first determine the attraction on a particle at the distance R from the centre, the direction of R making with the polar axis an angle whose sine is & CLA1RAUT. 219 By Clairaut's proposition in Art. 333, we substitute instead of the oblatum, a certain ellipsoid of revolution of equal mass. Let c denote the polar semiaxis, and \ the ellipticity of this derived ellipsoid. The attraction which it exerts mass of oblatum this may be deduced from Art. 261, supposing X positive ; or it may be obtained in Clairaut's manner, to which we have referred in Art. 334, and then it will be found to hold whether \ be posi- tive or negative. Now to our order of approximation c = r (I + es 2 ) ; and the condition of equal masses gives r 5 (1 + 2e) = c 3 (1 + 2\) = r 3 (1 + 3es 2 )(l + 2\) ; so that -('-) Also, supposing the attracted particle to be on a concentric and similarly situated oblatum, the dimensions of which are given by and e lt we have 4-Trr 3 The mass of the oblatum = ^ (1 + 2e) p. Hence the attraction of the oblatum = g; 3 (1 + 26) (1 - 2e/) p jl - gj (1 - | f) e} 9esV) Let us denote this for a moment by pf(r] ; then for the attraction of the shell of density p, comprised between the surfaces which correspond to r and r + dr> we have p-*^ dr. 220 CLAIRAUT. Hence, if we suppose the density of each shell to be uniform, but the density to vary from shell to shell, we have for the whole attraction *>> P J j J dr. o dr Let A I pr*dr t B \ p -j - dr, D = I p -j - c?r, then Jo . J $ ar J o dr the attraction is We must now introduce the centrifugal force. The centri- fugal force at the equator is approximately ^ ; and therefore it is ^ at the point under consideration : the resolved part of this along the radius is -- % , which must be subtracted from r i the attraction to obtain the gravity. Hence the gravity at the point under consideration 4?r 4- SjrB 4<7r (z w\ n 4f7rs 'J A 'rt-r-V** * Let P denote the gravity at the pole, g the gravity at the point under consideration; then Thus P g varies as s 2 ; that is, the diminution of gravity in passing from the pole to the equator varies as the "Square of the cosine of the latitude. Let E denote the gravity at the equator ; then P E = 4?r Divide this by E; then on the right-hand side it will be 4t7rA. sufficient to use 3- for E, so that P-E 3D CLAIRAUT. 221 Substitute for -r from Art. 327, and we have P-E 5 . This remarkable result is called Clairaut's Theorem. The frac- T) _ -p j-r- we shall call Clairaut's fract shall denote it by v; so that we have T) _ -p tion -r- we shall call Clairaut's fraction, as in Art. 171, and We know by Art. 28 that -j is twice the ellipticity of the earth, supposed homogeneous; and this is the form in which Clairaut himself expresses this term. 337. The assumptions on which Clairaut's demonstration of his famous theorem rests should be carefully noticed. The strata are supposed to be ellipsoidal, and of revolution round a common axis, and nearly spherical. Each stratum is homogeneous, but there is no limitation on the law by which the density varies from stratum to stratum : the density may change discontinuously if we please. It is not assumed that the strata were originally fluid; but it is assumed that the superficial stratum has the same form as if it were fluid and in relative equilibrium when rotating with uniform angular velocity. There is no limitation on the law by which the ellipticity varies from stratum to stratum, except that the ellipticity must be continuous, and at the surface must be such as would correspond to the relative equilibrium of a film of rotating fluid. We shall find that D'Alernbert in 1756 mistook the range of Clairaut's demonstration: see the Recherches sur. . . Systdme du Monde, Vol. in. page 187. In some modern works there has been a want of strict accu- racy as to the Theorem, owing perhaps to an undue regard to brevity. Thus we read in one that Clairaut established his Theorem on " the hypothesis of the Earth being a fluid mass " ; and we read in another that Clairaut discovered, his Theorem for " the case of a rotating fluid mass, or solid with density distri- buted as if fluid." 222 CLAIRAUT. 338. Clairaut on his pages 251... 259 uses his theorem to sup- port certain criticisms on Newton, David Gregory, and Maclaurin. We have already noticed these criticisms: see Arts. 30, 84, and 271. On his pages 260... 262 Clairaut in like manner uses his theorem in relation to Cassini's hypothesis that the earth was an oblongum with an ellipticity of ^ . In Art. 336 put l = - y o y o and * = 289 ; then We get a PP roximatel y u = 93 + 115 = 51 ' But, as Clairaut observes, this is a far greater value of v than pendulum observations warrant. Cassini's number however seems to have been 95 not 93 : see Art. 104. 339. Clairaut's fourth Chapter occupies pages 262... 296; it considers the figure of the Earth, supposed to have been originally fluid, and composed of strata of varying densities. In fact Clair- aut now proposes to investigate the connection between the density and the ellipticity in order that strata of the kind considered in the preceding Chapter may be in relative equilibrium if they are fluid. A process like that of Art. 823 must now be applied to each stratum. 340. Suppose a shell of density p bounded by two concentric and similarly situated oblata ; let f t be the ellipticity of the inner surface, and the ellipticity of the outer surface. Suppose a particle situated on the inner surface of the shell ; we shall deter- mine the attraction which the shell exerts on this particle in the direction at right angles to the radius vector from the centre of the shell to the particle. This problem is solved by Clairaut on his pages 262... 265. Our solution is substantially the same. The attraction of the shell is of course equal to the difference of the attractions of the oblatum which is bounded by the outer surface, and the oblatum which is bounded by the inner surface. We will consider these bodies separately, beginning with the larger. The larger oblatum produces the same effect as would be pro- duced by a similar, similarly situated, and concentric oblatum, having the particle on its surface ; for the difference of these two similar, similarly situated, and concentric oblata produces no effect by Art. 13. CLAIRAUT. 223 Hence the attraction of the larger oblatum in the assigned direction is ^ i n the notation of Art. 321 ; for now 7 = 0. o And, as in Art. 323, we have tan fS = 2 sin X cos X, so that the SirpcL sin X cos X attraction becomes - - ., - . In like manner the attraction of the smaller oblatum in the , ,. . . 8-Trpct sin X cos X assigned direction is - iT~ Thus it follows that the required attraction of the shell in the . 87rpc(> ) sin X cos X assigned direction is g . Now suppose that there is an infmitesimally thin shell of the density p ; let r be the polar semiaxis of the inner surface, and e the ellipticity of this surface ; then e 4- -r- dr will denote the ellip- Ctf ticity of the outer surface. Therefore the attraction, in the direc- tion at right angles to the radius vector, of this shell on the inside particle is J- c sin X cos X -r- dr ; this is obvious from the preced- O dr ing investigation. 341. We now proceed to apply a process like that of Art. 323 to any stratum. Let there be a particle of fluid in any stratum at the distance r from the centre ; let X be the angle between the radius vector to the particle and the polar semiaxis. 4-77- fr The central attraction on the particle is -, 2 1 pr*dr, approxi- r J o mately; for the strata beyond the particle produce no central effect to the order of accuracy which we have to consider. This central attraction gives rise to a component in the meridian plane, at right angles to the radius vector, towards the pole equal to 4nr[ r ' 2e' sin X cos X x ^ I pr'dr. We will call this a transverse at- T J o traction. 224 CLAIRAUT. The transverse attraction on the particle from the strata below ,. , . 8?rr' sin \ cos X [ r ' d . . N the particle towards the equator, is - ^-^ I p -=- (rejdr, by Art. 323. The transverse attraction on the particle from the strata be- ,. , , , . STI-/ sinXcos X f r i de 1 yond the particle towards the equator is ^ - I p dr, by Art. 340. Let CD denote the angular velocity ; then the transverse com- ponent of the centrifugal force is o/V sin X cos A,. Hence, as in Art. 323, equating to zero the whole transverse force, and dividing by 4-Tr sin X cos X we obtain 2e'f r ' 2 , 2 f d, BN , 2r l fi de , rV pr*dr - J p -j- (er 5 ) dr - -- p -y- dr - T - = 0. r J 5r J r dr^ o J r - dr 4?r If as usual we denote by J the ratio of the centrifugal force at the equator at the surface of the fluid to the attraction there, we have to the order of accuracy which we require ATT Substituting this value of w 2 our equation becomes 26'f 2 f^ A f 5 , , 2r[* de jr [* 2 , pr*dr -^-TI p -j- (er 5 ) dr - \ p -=- dr -*] prdr = 0. r' 2 J ' 5r /4 J r dr v o J r > ^ dr r^Jo This important equation occurs for the first time in Clairaut's page 273 ; it has ever since been permanently associated with the subject : I shall call it Clairaut's primary equation. Whether we leave o> 2 in the equation, or substitute for it in the manner just explained, is of no consequence. 342. If and p are taken so as to satisfy Clairaut's primary equation we have a possible constitution for the earth. Clairaut however asserts more than this on his page 265, namely that if j be very small the strata will be elliptical spheroids. Even Laplace has scarcely arrived at this point ; he has only shewn that if the strata are assumed to be nearly spherical they must be elliptical spheroids. CLAIRAUT. 225 343. Clairaut transforms his primary equation. It will not lead to any confusion if we now drop the accent from r and from e' : we may then write the equation thus : f VVr - 2 f "p * K) dr - to* I"' P % dr - *$? = 0. Jo Jo r dr Jr dr 47r Differentiate with respect to r ; thus Op/e - 2p ^ (er 6 ) 4 [ n de , de 25rV -10r 4 p^ Jr ^ dr -j dr 4?r Simplify, and divide by 10r 4 ; thus ide 2e\ r , * de Differentiate again with respect to r ; thus so that -j^+ ry = / - 2 dr C r , I r dr 6 2pr \ \e. This I shall call ClairauCs derived equation. 344. Clairaut puts his derived equation in another form. Let - -j- be denoted by u : so that ^ = eu. edr dr mr e 2 Then _ =eM2 + 6 _. Thus f T. M. A. 15 2pr 226 CLAIRAUT. Put u + 7 - = t ; and then we obtain Clairaut observes that this equation falls under the case of dy + y n dx = Xdx, where X is a function of x ; and that what we now call the separation of the variables had not yet been effected in general. Accordingly he does not propose to seek for the ellipticities which correspond to a given law of density, except in the case in which p varies as r n . See his page 276. 345. Suppose then that p varies as r n . We have by the preceding Article dt , 2 n* + 3/i + 6 -T- + * = - jj - . dr r* This becomes homogeneous and easily integrable by putting a new variable instead of - ; and thus we obtain r where q = I ( n* + 3ra + -7- J , and a is an arbitrary constant. With this value of t we find where b is another arbitrary constant. This value of e then satisfies Clairaut's derived equation ; we must examine if it also satisfies the primary equation. Substitute the value of e, and we find after simplification that the primary equation is satisfied provided the following relation holds between the constants : (? - g - ) ftr,* - ba (q + I + n) rH = f r. CLAIRAUT. 227 Thus there is only one relation between two constants, and so it would appear that the solution is not determinate. Clairaut offers an explanation on this point. It has been assumed through- out that the ellipticity of the strata is small ; moreover he considers that n must be negative in order that the density may diminish from the centre to the circumference, which he says the laws of hydrostatics require : see Art. 315. Hence we must have a = ; for otherwise e would be infinite at the centre. Also even if n be considered positive we must have a = ; in this case e would be finite at the centre, but re would be infini- tesimal : see Clairaut's page 281. For a particular case suppose n 0, then q = - ; and after A putting a = we obtain b = - : and then e = b = -j as it should be. 4 4 346. Clairaut's derived equation may be put in the form ,#* 6e r -T-5 a dr* pr 2 Then if be given as a function of r, the left-hand member of the equation becomes a known function of r ; denote it by P\ from this we deduce pr* = Pe^ '* which gives p. See Clairaut's page 283. 347. The formulae which have been investigated for the case of an infinite number of indefinitely thin strata may be applied to the case of a finite number of shells surrounding a central part, the density changing abruptly from the central part to the adjacent shell, and then from shell to shell. Clairaut considers this on his pages 286. . .293 ; taking the density constant throughout the central part, and throughout each shell. He shews that the ellipticities increase from the centre to the surface, assuming that the densities diminish from the centre to the surface : see his page 292. 152 228 CLA1RAUT. Clairaut takes for an example the case in which the whole mass is supposed to consist of two parts throughout each of which the density is constant. Let p l be the density of the outer part, j the ellipticity of the outer surface, r, the polar semiaxis ; let p 2 , e 2 , and r 2 be corresponding quantities for the inner part. Then in the integrations which occur in Clairaut's primary equation we have to make p = p 2 from r = to r = r 2 , and p = p l from r = r 2 to 7* r = TI . Put X for , and suppose p 2 = p t + cr. Then apply Clairaut's r g primary equation first to the extreme stratum of the inner part, and next to the extreme stratum of the outer part : thus we obtain after reductions e t (10ft + 4cr) 66^ = 5j (p l H i (^ft + 10(7\ 3 ) 6e 2 o-X 5 = 5/ (PJ From these equations we deduce 2 3 (10ft + 4(7) (2ft- 348. In his pages 294 and 295, Clairaut points out two limits for the ellipticity of a planet, assuming that the planet was origi- nally fluid, and that the denser strata are the nearer to the centre. One limit is the ellipticity which corresponds to the case of a homogeneous mass. The other limit is that in which the attraction at any point is directed towards the centre, and varies inversely as the square of the distance from the centre, for this may be regarded as equivalent to having the density infinite at the centre: see Arts. 64 and 173. Clairaut states on his page 296 that the theorem which we now call Clairaut's Theorem, holds for the case in which the earth is supposed to have been originally a fluid of the nature which he has considered. This is obvious from the demonstration already given : see Art. 336. CLAIRAUT. 229 349. Clairaut's fifth Chapter occupies pages 296. ..304; it is OD the comparison of the theory with observation. Clairaut considers that the observations of the diminution of gravity in passing from the pole to the equator agree sufficiently well with his theory. But the comparison of the French and Lapland arcs gave the ellipticity apparently greater than whereas his theory required the ellipticity to be less than r . But, as he justly says, the comparison of these arcs was sufficient to establish the oblateness of the earth, but not to determine accurately the ratio of the axes ; for the latter purpose the measurement of more distant degrees was required. He alludes to the operations in Peru, the result of which was now expected ; this became known soon after the publication of Clairaut's work. The comparison of the French and Peruvian arcs would have given a smaller ellipticity, and therefore more favourable to Clairaut's Theory: see Boscovich De Litter arid Expeditions, page 501. Some years later Clairaut made an attempt to explain the conflict between theory and observation as to the Figure of the Earth in an Essay which received a prize from the Academy of Toulouse ; but this Essay seems never to have attracted any atten- tion : I shall give some notice of it in Chapter XV. 350. Clairaut's work is one of the most interesting and remarkable in the literature of mixed mathematics. Laplace says in the Mecanique Celeste, Vol. V. page 7, after an analysis of the work: I/importance de tous ces re"sultats et Pelegance avec laquelle ils sont present6s, placenfc cet ouvrage au rang des plus belles productions mathematiques. In the Figure of the Earth no other person has accomplished so much as Clairaut ; and the subject remains at present substan- tially as he left it, though the form is different. The splendid analysis which Laplace supplied adorned, but did not really alter, the theory which started from the creative hands of Clairaut. 230 CLAIRAUT. Physical astronomy began with Newton in England ; the memoirs which Maupertuis and Clairaut contributed to the Philo- sophical Transactions may be regarded as a graceful tribute to the country which gave birth to the greatest of scientific men. Newton, according to Bailly, reigned alone; but at his death, his empire, like that of Alexander, was divided : and Clairaut, D'Alembert and Euler succeeded. Histoire de V Astronomic Moderne, Vol. HI. page 154. Perhaps the names of Maclaurin and of Thomas Simpson ought to be recorded among the successors of Newton, but I fear it cannot be denied that on the whole his countrymen have left to foreigners the glory of continuing and extending his empire. England has produced numerous patient and able observers, but for the modern theory of physical astronomy we must chiefly study the great French writers, including among them two Italians, Lagrange and Plana, who in language have associated themselves with Laplace. CHAPTER XII. ARC OF THE MERIDIAN MEASURED IN PERU. 351. WE have seen in Chapter VII. that the expedition for measuring an arc of the meridian in Lapland started from Paris after that which went to Peru ; nevertheless, the question as to the oblate or oblong form of the Earth was settled by the Arctic company before any result had been obtained at the Equator. In accordance with the plan of the present work, we might, therefore, leave the operations in Peru without further notice; but their extent and importance will justify us in devoting some space to a brief sketch of their course and conclusion. 352. It will be convenient to collect together the titles of the original works, accompanied with an indication of the nature of their contents. They will be arranged in the order of publication, and distinguished by Roman numerals, for the sake of easy reference. I. La Condamine. Relation abrdgte d'un Voyage fait dans Vinterieur de TAmdrique Mfaidionale. 8vo. Paris, 1745. This gives an account of the voyage which La Condamine made down the river Amazon on his return home ; it is a very interest- ing volume, but does not relate to our subject. II. A Spanish translation of I., or of part of I., with some additions, seems to have been published at Amsterdam in 1745 : see a note on page x. of XIII. ; and also the life of La Condamine, by Biot, in the Biographic Universelle, republished in Biot's M6- langes Scientifiques et LitUrcdres, Vol. III. 232 ARC OF THE MERIDIAN MEASURED IN PERU. III. La Condamine. Lettre...sur VEmeute populaire excitee en la Ville de Cuenqa This seems to have been published at Paris in 1746 in octavo. It contains an account of a tumult at Cuen9a in 1739, which led to the death of Seniergues the surgeon of the French expedition. La Condamine encountered great trouble in carrying on the prose- cution of the guilty persons. IV. An English translation of I. was published at London in 1747. According to Biot, cited under II., there was also a Dutch translation. V. Bouguer. In the Paris Mdmoires for 1744, published in 1748, there is a memoir entitled Relation abregte du Voyage fait au Pe'rou.... The memoir occupies pages 24.9. .297 of the volume ; it was read on the 14th of November, 1744. There is an account of the memoir on pages 35... 40 of the historical portion of the volume. . The memoir consists of two parts. The first part relates to the voyage ; and this is an abridgement of the introductory por- tion of IX. The second part is an outline of the operations described at full in the body of IX. Bouguer is rather rash on his page 296 ; he made some obser- vations with a common quadrant in 1738, and says : " je vis assez clairement que 1'aplatissement alloit aussi loin que 1'a pretendu ce grand homme [Newton]"... Thus he saw clearly what we now know did not exist. The passage does not appear to be repro- duced in IX. La Condamine was more cautious than Bouguer as to this matter. XIII. 63, XVIII. 64. VI. Juan and Ulloa. Relation Historica del Viage a la America Meridional.. A vols. 4to. Madrid, 1748. The first volume contains pages 1...404, besides Half-title, Frontispiece, Title, Preface, and Table of Contents. The second volume contains pages 405... 682, besides Half-title and Title. The third volume contains pages 1...379, besides Half-title, Frontispiece, Title, Table of Contents, and Errata. The fourth volume contains ARC OF THE MERIDIAN MEASURED IN PERU. 233 pages 380... 603 and i...cxcv, besides Half-title and Title. In the first and second volumes there are plates and maps, which are numbered from i. to xxi. continuously. In the third and fourth volumes there are plates and maps, which are numbered from i. to xii. continuously ; and also a sheet containing the portraits of twenty-two emperors of Peru, beginning with Manco-Capac, the fabled child of the sun, and ending with Ferdinand the Sixth of Spain. These four volumes give the account of the occupations of the two Spanish officers, and a description of the countries of Peru and Chili and of their inhabitants ; they were drawn up by Ulloa. They form an interesting work, which, however, is very slightly connected with our subject. VII. Juan and Ulloa. Observaciones Astronomicas, y Phisi- COS...MO. Madrid, 1748. Pp. xxviii+396, besides Half-title, Frontispiece, Title, Preface, Table of Contents and Index. There are plates numbered continuously from i. to viii. ; besides a map of the moon. This volume contains the detail of the geodetical and astronomical work, drawn up by Juan ; it is an essential adjunct to VI., though copies of VI. are sometimes found without VII. We will return to this volume : see Art. 362. VIII. La Condamine. In the Paris Memoires for 1745, pub- lished in 1749, there is a memoir entitled Relation abrtgte d'un voyage fait dans linterieur de TAmtrique Mtridionale... The memoir occupies pages 391... 492 of the volume: it was read on the 28th of April, 1745. There is an account of the memoir on pages 63... 73 of the historical portion of the volume. This memoir agrees substantially with I. ; but the two are not identical. A few passages occur in the memoir which are not in the book. Perhaps the book, which was published first, coincides with the discourse actually read to the Academy ; and then the memoir received the slight additions before the volume for 1745 appeared. The memoir contains a plate which is not given in the book. This consists of a chart and a view of a remarkable part of the Amazon, where the river runs in a narrow channel between high rocks. 234 ARC OF THE MERIDIAN MEASURED IN PERU. IX. Bouguer. La Figure de la Terre... 4to. Paris, 1749. Pp. ex + 394, besides Title, Avertissement, Table, and Errata. This is the most elaborate work for our purpose to which the expedition gave rise ; we will return to it : see Art. 363. X. Bouguer. In the Paris Memoires for 1746, published in 1751, there is a memoir entitled Suite de la Relation abregte, donntfe en 1744,... The memoir occupies pages 569... 606 of the volume : it was read on the 18th of February, 1750. This contains the geodetical measurements and the astrono- mical observations : it is an abridgement of the corresponding part of IX. to which Bouguer refers for full information. XI. La Condamine. In the Paris Memoires for 1746, pub- lished in 1751, there is a memoir entitled Extrait des operations Trigonome'triques, et des observations Astronomiques... The me- moir occupies pages 618... 688 of the volume: it was read on the 27th of May, 1750. This is an abridgement of XII., which was just about to be published. La Condamine says : " J'ai use du droit d'auteur en faisant mon extrait, et on y trouvera quelques particularity's omises dans le livre meme." These additions to XII., however, are small and not important. XII. La Condamine. Mesure des trois premiers degrts du Mtridien... 4to. Paris, 1751. Pp. 266 + x, besides Title, Avertisse- ment, and Table. This is La Condamine's account of the scientific operations. It is divided into two parts ; the first part relates to the geode- tical measurements, and the second to the astronomical observa- tions. The pages 239... 258 contain an important discussion of Picard's operations. XIII. La Condamine. Journal du Voyage fait par ordre du Roi... 4to. Paris, 1751. Pp. xxxvi + 280 + xv, besides Title. This is La Condamine's account of the voyage and the resi- dence in Peru. XIV. A French translation of VI. and VII. was published at Amsterdam and Leipsic in 1752, 2 vols. 4to. The first volume ARC OF THE MERIDIAN MEASURED IN PERU. 235 contains Frontispiece, Title, Dedication, Publisher's Advertisement, Preface, Table of Contents, and Errata, and then 554 pages of text. The second volume contains Frontispiece, Title, and Table, and then 316 pages of text, with an index for the history of Peru. This brings us to the end of the translation of VI. ; and the remainder of the volume is devoted to VII. : this consists of Title, Preface, Table of Contents, 309 pages of text, and an Index. The transla- tion has the same plates and maps as the original, except the sheet with the portraits of the emperors of Peru. The translation has in addition plans of Cape Francois and of Louisbourg ; and also eight plates which are intended to illustrate the early history of Peru. We learn from the Publisher's Advertisement that this trans- lation was not allowed to be published at Paris. The translation of VII. is very unsatisfactory ; many passages are here perverted into absolute nonsense, which are quite in- telligible in the Spanish original. XV. There is an English translation of VI. I have not seen any edition except the third, which is dated 1772, and was pub- lished at London. This is in two octavo volumes. The first contains xxiv + 479 pages; the second contains 419 pages, besides the Title, Contents, and Index. There are plates and maps which are numbered from i. to vii. continuously ; these reproduce on a small scale most of the illustrations of the original work. The English translation omits the following portions of the original : the explanation of the construction and use of the sextant, Vol. I. pages 196. ..213 ; the description of the map of the western coast of South America, Vol. IV. pages 469... 485; and the sketch of Peruvian history, Vol. IV. pages L..CXCV. Moreover, Ulloa in returning to Spain was taken prisoner by the English ; and he complains of the barbarous treatment he received from those who captured him, Vol. IV. pages 447 and 517 : these com- plaints are omitted in the English translation. XVI. Bouguer. Justification de plusieurs faits. . . 4to. Paris, 1752. Pp. viii + 54, besides a double Title, and a leaf containing the Approbation, Privilege du Roi, and Errata. 236 ARC OF THE MERIDIAN MEASURED IN PERU. This is an attack on La Condamine; it is of no scientific value, for it does not bear on any of the results obtained by the expedition, but only on trifling personal matters. For example, Bouguer's first twenty-one pages are spent on maintaining that the other Academicians were disposed to begin by measuring an arc of the Equator, before the orders from France were received which required them to confine themselves to an arc of the meridian. Even if Bouguer established this point, which is not certain, there cannot be any importance attached to it. XVII. La Condamine. Supplement au Journal Historique... Premiere Partie. 4to. Paris, 1752. Pp. viii+52, besides the Title and Approbation. XVIII. La Condamine. Supplement au Journal Historique ... Seconde Partie. 4 to. Paris, 1754. Pp. 222 + xxviii, besides the Title, Avertissement and Approbation. There are also two pages containing supplements to the Errata for XII. and XIII. In XVII. and XVIII. we have the reply of La Condamine to XVI. XIX. Bouguer. Lettre ... divers points d Astronomic pra- tique... 4to. Paris, 1754. Pp. 51 besides the Title and Approba- tion. This is a rejoinder to XVII. and XVIII. XX. Reponse de Monsieur * * * d la Lettre de M. Bouguer, sur divers points ... Pp. 11. I have seen only one copy of this publication ; and that had no indication of date or place. It contains a page by an anony- mous writer, which introduces a letter from La Condamine, con- stituting a rejoinder to XIX. XXI. La Condamine. Relation abrfyee... 8vo. Maestricht, 1778. Pp. xvi + 379, besides Title and Approbation. This consists of a reprint of I. and II., augmented by two letters. One letter is from La Condamine, and contains a sketch of the fortunes of the members of the arctic and equatorial expeditions up to about 1773. The other letter is from one of the subordinate members of the equatorial expedition, Godin des ARC OF THE MERIDIAN MEASURED IN PERU. 237 Odonais ; it gives an account of the calamities which befell his wife on her return to Europe down the Amazon. In the Quarterly. Beview, Vol. 57, 1836, will be found a description of two modern voyages down the Amazon by English explorers, and also some notice of the sufferings of Madame Godin des Odonais. XXII. There is a reprint of XVI. also in 4to. Paris, 1809. This is in rather smaller type than the original, and contains vi -f- 44 pages, besides the Title. I am at a loss to imagine what could have been the motive for reprinting this controversial piece so many years after all the persons concerned had passed away. 353. \Ve will now give a brief account of the operations of the expedition, and the results obtained ; we shall cite the pages of the original works from which our statements are derived. The French expedition left Rochelle on the 16th of May, 1735, and arrived at Carthagena on the 16th of November; the two Spanish officers had already been waiting there for several months. The party reached Panama on the 29th of December. XIII. 3, 5, 8. A base was measured during October, 1 736, near Quito ; the whole party was divided into two bands : one band measured from the north end to the south, and the other from the south end to the north. The difference between the two measurements was less than three inches in 6273 toises. XII. 5. The geodetical angles were observed with quadrants. La Condamine's quadrant had a radius of three feet, Bouguer's about two feet and a half, Godin's not quite two feet; the Spanish officers, after their arrival in Peru, received from Paris a quadrant intermediate in size between Godin's and Bouguer's. IX. 60, XII. 13. There were two series of triangles ; one was measured by Godin and Juan ; the other was measured by Bouguer and Ulloa, and also by La Condamine. The two series, had about half their triangles common ; differing only towards the extremities of the arc. Thus three separate Trigonometrical measurements were obtained, each of which may be considered complete and inde- pendent. Every angle was observed; each person observing at least two angles of every triangle. XII. 12... 15. 238 ARC OF THE MERIDIAN MEASURED IN PERU. 354. The geodetical work was carried on with great difficulty owing to the nature of the country. There was a narrow valley running nearly in the direction of the meridian, between two lofty chains of mountains. On the elevated points, which were chosen for stations, the observers suffered much from the inclemency of the weather ; and they were often compelled to remain for several days or even weeks, to obtain a glimpse of the points for which they were looking, as these points were usually enveloped in mist. XIII. 52. More than once a report was current that the ob- servers had perished. On the occasion of one very severe storm, to which they were exposed, public prayers were offered for them ; or as La Condamine cautiously adds, " du moins on nous Tassura." VI first Vol. 314, XIII. 81. The Indians caused much trouble by deserting in critical circumstances, and by incurable dis- honesty. XIII. 50, 52, 72. The upper classes, who were of course Spaniards, at least by descent, seem to have received the expedition in general with politeness and kindness. XIII. 65, 75. But on the other hand we must place the tumult excited at Cuen9a, by which the French surgeon lost his life. Moreover, a frivolous charge of acting contrary to the orders of the king of Spain, was on one occasion brought against La Condamine ; and on another occasion he was disturbed by a nocturnal visit of a police official. XIII. 26... 30, 101. La Condamine seems to have had great trouble and anxiety respecting matters which ought not to have been thrown on a person fully occupied with his proper scientific work. He had at the commencement of the operations to un- dertake a voyage to Lima, in order to procure money for the expenses. XIII. 19... 25. He had to engage in tedious proceed- ings at law in order to prosecute the persons who had caused the death of the surgeon. XIII. 86. He was also involved in a vexatious business connected with the erection of two pyra- mids to mark the extremities of the measured base, and with the inscriptions to be placed on them: these pyramids were finally destroyed by orders from Spain. La Condamine devotes a large space to the history of the pyramids, prefixing the motto " Etiwni peridre ruince." XIII. 21 9,.. 280. According to the Avertissement to XIV. Ulloa was about to issue a history of the transactions rela- tive to the pyramids : I do not know whether this ever appeared. ARC OF THE MERIDIAN MEASURED IN PERU. 239 355. Near the South end of the arc a base of verification was measured. Bouguer and Ulloa measured it from South to North ; La Condamine and Verguin, the draughtsman to the expedition, measured it from North to South. The two measures agreed within two inches in 5259 toises. Part of this base was measured across a shallow pool ; the measuring rods floated on the surface. The calculated length of the base of verification differed from the measured length by about a toise. XII. 72, 85 ; XIII. 83. Godin and Juan also measured a base of verification, near the South end of the arc, but not the same as that just noticed. VII. 165 ; XIII. 83. 356. The astronomical part of the operations was naturally the most difficult and the most important: we must now for a time fix our attention on Bouguer and La Condamine. Any sketch will give but a faint idea of the obstacles which had to be over- come, and of the assiduity of the observers ; and, indeed, there is danger lest a sketch should contain or suggest some erroneous notions. The star e of Orion was selected for observation at both ends of the arc. At the North end this star crossed the meridian to the South of the zenith, and at the South end it crossed the meridian to the North of the zenith. Thus the two zenith distances had to be found, and their sum gave the amplitude of the arc. A sector of 12 feet radius, with an arc of 30, had been brought from France; this was used in some observations for determining the obliquity of the ecliptic at the early part of the residence in Peru. But the arc was far longer than was neces- sary for the zenith distance of the selected star ; and so Bouguer and La Condamine substituted a new arc. The most remarkable circumstance connected with the new arc is, that it was not graduated. The zenith distance of the star was known approx- imately; an are was taken nearly equal to this known value, and having its chord a certain submultiple of the radius : this arc was set off on the limb of the instrument. Then the differ- ence between this arc and the actual zenith distance of the star was determined by the aid of a micrometer. XII. 108. Some dispute 240 ARC OF THE MERIDIAN MEASURED IN PERU. arose afterwards as to the person to whom the credit of this con- trivance was due. XII. 120; XVI. 36; XVIII. 111. 357. Observations were made by Bouguer and La Condamine at Tarqui, the southern station, in December, 1739, and January, 1740 ; and at Cotchesqui, the northern station, in February, March, and April, 1740. They appear to have been at the time contented with their results, and to have considered the object of the expedition fulfilled. XII. 165. I do not perceive any distinct statement of the causes which led Bouguer and La Condamine to suspect the accuracy of the astronomical observations of 1739 and 1740, and in consequence to postpone their return to Europe. Perhaps La Condamine was detained by the affairs of the death of the French surgeon, and of the pyramids. They naturally wished before they left Peru to compare their result with Godin's; and Godin had not yet ar- rived at his conclusions. XIII. 105. However, Bouguer made more observations at Tarqui ; and towards the end of 1741 he announced to La Condamine that the work which they had imagined to have been finished more than a year since must still be continued for several months : the old observations at Tarqui were to be rejected because they differed so much from the more recent observations. XIII. 128. The untrustworthiness of the early observations seems to have been due mainly to a want of rigidity in the whole instrument, composed of radius, limb, and telescope. One unfortunate circum- stance, for example, was that the radius had been constructed in two pieces for facility of transport from France; and when the instrument was to be used, the screws could not be found which were to fasten the two parts firmly together. XVIII. 42. The necessary rigidity was finally secured by the aid of strengthening bars and wire. But even after his return to France La Condamine considered that the matter was not fully explained. XVIII. 73. It is obvious also that the optical defects of the telescope gave great trouble. The single object-glass could not bring all the different coloured rays to the same focus ; and thus in the use of the micrometer there was an opening for serious error. Both ARC OF THE MERIDIAN MEASURED IN PERU. 241 Bougtier and La Condamine treat at length on this matter, but not with perfect clearness. IX. 202. ..214; XII. 196. ..215. 358. Finally, simultaneous observations of the star were made by La Condamine at Tarqui and by Bouguer at Cotchesqui, towards the end of 1742 and the beginning of 1743. Those by Bouguer were made with a new sector of 8 feet radius, con- structed under his own direction and inspection. Those by La Condamine were made with the 12 feet sector, improved successively by Bouguer and himself. XII. 185, 190. By taking simultaneous observations, the corrections for precession, nutation, and aberration, were rendered unnecessary. The aberration of light was known, but not the laws of the correction which it involved for observations of the stars. XII. 139, 220. The amplitude of the arc was found to be about 3 7' 1". La Condamine obtains 56749 toises for the length of the first degree of the meridian reduced to the level of the sea. XII. 229. Bouguer gives 56753 toises. IX. 275. Delambre recalculated the astronomical work of Bouguer and La Condamine ; and fixed the amplitude at 3 1' 3". He took a mean between the lengths assigned by Bouguer and La Condamine, and thus obtained, for the length of "the arc reduced to the level of the sea, 176877 toises. See Base du Systeme Metrique,... Vol. ill. page 133. The corresponding length of a degree is about 56737 toises. 359. We stated at the commencement of Article 356 that we confined ourselves to the proceedings of Bouguer and La Con- damine. Let us now advert to the other members of the party. Godin himself published no account of his operations ; nor have I ever seen any reference to manuscripts which he may have left. Much of his work, however, was executed in association with Juan; and there is good reason to conclude that his re- sults must have agreed substantially with those of Bouguer and La Condamine. XII. 231 ; XIII. 140. The arc on which the Spanish result depends fell rather short of the arc of Bouguer and La Condamine at the southern end, but wcrnt beyond it at the northern end. The extension of the arc northwards introduced five new triangles ; Juan and TJlloa were T. M. A. 16 242 ARC OF THE MERIDIAN MEASURED IN PERU. both concerned in this extension, and I presume that Godin also was with them. VIL 167, 224 ; XIL 231. The details connected with the triangles as observed and calculated both by Juan and Ulloa are recorded. VII. 144, 214. For the astronomical work Godin constructed a very large sector; this is said in various places to have had a radius of 20 feet : but La Condamine correcting his former statements put it ultimately at 18 feet. VII. 272 ; IX. 273 ; XIII. 85, 99 ; XVI. 38 ; XVIII. 77. Observations of three stars, e of Orion, 6 of Antinous, and a of Aquarius were made at Cuenc,a, the southern station, in August and September, 1740, by Godin, Juan, and Ulloa. The Spanish observers were then withdrawn from their scientific occupa- tions, and employed in the naval service, to assist in defending the country against the expected attacks of the English. Hence the observations at Pueblo Viejo, the northern station, were not made by them until April and May, 1744; Godin did not assist at these. VII. 283. The amplitude of the arc was finally settled at 3 26' 52f". We do not see in the Spanish account anything corresponding to the excessive trouble which Bouguer and La Condamine ex- perienced in their astronomical observations ; we learn little more than this, that the first large sector which was made was unsatis- factory, and so another was made. VII. 271. The Spanish result gave 56768 toises for the length of the degree of the meridian. VII. 295 ; XII. 234. 360. Bouguer arrived in Paris towards the end of June, 1744, about eight months before La Condamine. XIII 215. A violent controversy subsequently arose between them ; and this leads us to enquire on what terms the Academicians had been during their operations. Godin separated himself from the other two in Peru. XVIII. 43. Bouguer seems to have been displeased at this, but La Condamine does not record any disapprobation. IX. 228 ; XII. 106 : see also XVIII. 6. La Condamine asserts that he had been on good terms with Bouguer during the ten years of the expedition, and for three ARC OF THE MERIDIAN MEASURED IN PERU, 243 years afterwards. XVII. iii : see also XVII. 28, 30 ; XVIII. 180, 203, 206. But on the other hand there are statements which imply that there must have been a want of perfect cordiality between these two, even in Peru. XVIII. 6, 62, 64, 143, 175, 182; XIX. 18, 49. Each of them claims to have been on good terms with Godin. XVI. 39 ; XVIII. 43 ; XIX. 38. The date of the public explosion is November 1748 ; the cause was the charge made by Bouguer, that his colleagues were inclined to measure a degree of the equator, instead of a degree of the meridian, until arrested by orders from France. XVIII. 67, 212. The strife extends over the series of works XVI... . XX. ; but even these seem to have* formed but a small portion of the statements, verbal and written, which were brought before the Paris Academy. There was scarcely any exaggeration in La Condamine's complaint, that ten years of labour in the new world were followed by as many of controversy in the old. XVIII. iii, 190. The quarrel seems to me remarkable, alike for its fierceness and for the triviality of the matters in dispute. Thus, besides the measuring of an arc of the equator, to which we have already alluded, there is much conten- tion as to the origin and value of certain suggestions in optics and practical astronomy. A sketch of the history of the quarrel, followed by a summary of the main points, is given in XVIIL 205... 221. My own sympathy is on the side of La Condamine, although I consider Bouguer to have been by far the superior as a mathematician and an astronomer. 361. We may give a cursory notice to some miscellaneous points. The equatorial expedition was suggested by Godin ; see IX. iv, and Bailly's Histoire de I'Astronomie Moderne, Vol. in. page 11, note. Godin seems to have proposed it to the Paris Academy in 1734 ; but even in 1733 La Condamine had offered to measure degrees near the equator at Cayenne. IX. iv ; XVII. 28 ; XVIII. 190. When La Condamine, on his return home, arrived at Guyana, he came to the conclusion that the country was well adapted for trigonometrical operations. XXI. 188 ; XIII. 194, 201. And at a later period he bitterly regretted that his original design had not been carried out, and then he would not have 162 244 ARC OF THE MERIDIAN MEASURED IN PERU. lost the ten most precious years of "bis life in preparing vex- ations for ten more. XVIII. 190. Spherical trigonometry was now employed, apparently for the first time, in geodetical calculations ; this improvement is claimed by Bouguer. IX. 131 ; X. 584 ; VII. 255. To Bouguer is also due the idea of making observations with the view of determining the attraction of the mountain Chim- borazo ; La Condamine contributed a valuable suggestion in the practical operation. XVIII. 146. We shall now give more details respecting the works VII. and IX. 362. The Spanish volume of observations and experiments begins with a Preliminary Discourse, which consists of a history of opinions and investigations with respect to the Figure of the Earth. After having explained the views of Newton and Huygens, which involved the hypothesis of the rotation of the Earth, Juan says: Assi discurrian estos grand es ingenios en la Hypotesis del movimiento diurno de la Tierra ; pero aunque esta Hypothesis sea falsa, . . . The French translation supplies the following significant note : On doit se souvenir que TAuteur de cet Ouvrage, ne parle pas en Mathematicien quand il suppose faux le sentiment de ceux qui affirment que la Terre tourne, mais en Horn me qui ecrit en Espagne, c'est-a-dire dans un Pays ou il y a une Inquisition. The volume is divided into nine books, which treat on the following subjects : the obliquity of the ecliptic, observations of latitude, observations of longitude, expansion and contraction of metals, barometrical experiments, the velocity of sound, the length of the degree of the meridian, pendulum experiments, navigation on the surface of the oblatum. Juan holds that the Earth is an oblatum, and that the ano- malies which seem to occur may fairly be attributed to errors of observation. ARC OF THE MERIDIAN MEASURED IN PERU. 245 In order to obtain the ellipticity of the Earth, Juan assumes that in passing from the Pole to the Equator the seconds pendu- lum increases 2*16 lines. Hence by using Clairaut's theorem he obtains ? - for the ellipticity. See his page 334. The 216 lines is, I presume, an arbitrary value ; for although it would appear from his page 344, that this is in exact conformity with the observations of Maupertuis in Lapland, yet this must be a misprint, as we see by page 331. An investigation is given on pages 337... 345 for the rectifi- cation of the ellipse. Two infinite series are obtained, one for the length of an arc measured from the end of the minor axis, and the other for the length of an arc measured from the end of the major axis ; the former is nearly correct, the latter very much less so. The mathematical process is rather clumsy; for to expand - r in powers of #, Juan in effect expands (1 a? 2 ) 5 , (I-* 2 ) 2 and then divides unity by the series ; instead of simply expand- ing (1 # 2 ) '-. To ensure tolerably rapid convergency, Juan proposes to calculate the arc from the end of the minor axis up to a certain point by his first formula, and the arc from this point to end of the major axis by his second formula. However he finally in his numerical work retains only what we should call the square of the excentricity, and it is easy to see that to this order of accuracy he might have avoided infinite series altogether, and expressed his required result in a simple finite form. In treating on navigation Juan refers to a work by Murdoch, of which we shall give some account hereafter. Juan supplies tables of Meridional Parts, like Murdoch's, but much more copious, as they are calculated to every minute instead of to every degree. Juan adopts in his Tables - fc r the ellipticity. 3G3. Let us now turn to Bouguer's Figure de la Terre. The ex. preliminary pages give an account of the voyage and a description of the physical peculiarities of Peru, and the character of the inhabitants. 246 ARC OF THE MERIDIAN MEASURED IN PERU. The 394 pages of text are divided into seven sections. The first section is mainly devoted to shewing that it was advisable to determine the length of a degree of the meridian rather than the length of a degree of the equator. The second section gives an account of the triangles, including the measurement of the base. The third section treats of the reduction of the triangles to the plane of the horizon, and the determination of the situation of the sides with respect to the meridian. The fourth section relates to the precautions taken with re- epect to the astronomical observations. The fifth section contains the astronomical observations. The pages 227... 258, however, do not belong to our subject; they relate to the observations for determining the obliquity of the ecliptic which were made during the early part of the residence in Peru. The sixth section is thus entitled: Qui contient diverses re- cherches sur la Figure de la Terre et sur les proprietes de cette Figure. The investigations of this section are interesting, though rather speculative than practical. Bouguer considers the curve which represents the meridian of the Earth as unknown; but from this curve he supposes another deduced by the perpetual intersection of the normals, and he calls the deduced curve the gravicentrique : it is the evolute of the meridian curve in the language of modern mathe- matics. Bouguer investigates properties of the gravicentrique on the supposition that the length of it measured from the equator varies as the wi tb power of the sine of the latitude. He specially considers the cases in which m = 2, m = 3, and ra = 4 : see his pages 284... 289. The law for the length of the gravicentrique is also the law for the increase of the radius of curvature of the meridian in passing from the equator to the pole. The results of observation which had to be satisfied were the lengths of a degree of the meridian in Peru, France, and Lap- ARC OF THE MERIDIAN MEASURED IN PERU. 247 land Bouguer at first adopted the usual hypothesis of w = 2, and obtained - : for the ellipticity: see his page 297. But after the French degree had been corrected, this hypothesis did not seem to him to agree with the observations ; accordingly he supposed TTC = 4, and obtained ^^ for the ellipticity: see his page 303, Besides the three degrees of the meridian, he also pays attention to the degree of longitude which had been mea- sured towards the South of Franca Bouguer's hypothesis of m = 4 is quite arbitrary. It had, however, sufficient vitality to experience the adverse criticism of Laplace, who shews that it is inconsistent with pendulum obser- vations. Me'canique Celeste, Livre in. 33. Bouguer in his pages 319... 326 explains the nature of the changes which must be made in certain tables constructed for navigation, on the hypothesis that the Earth is spherical, in order to adjust them to the actual fact. Bouguer's seventh section is entitled Detail des Experiences ou Observations sur la gravitation, avec des remargues sur les causes de la Figure de la Terre. This section contains some very interesting matter, although there is nothing as to what we usually understand by the theory of the Figure of the Earth. Bouguer says on his page 327 : Nous n'entreprendrons point de nous clever jusqu'S, une Theorie complette de la Figure de la Terre ; parce que nous ne voulons rien dormer s'il est possible & nos conjectures. Bouguer describes the. way in which he made his pendulum experiments ; and then considers what reductions must be applied to the immediate results. He allows for the diminution of the weight of the pendulum caused by the air which it displaces ; he says that this correction is now made for the first time : see his page 340. He adverts to the effect of the resistance of the air ; and he states as a result which could be obtained by investiga- tion, that the time occupied in the ascending part of an oscillation will be diminished as much as the time occupied in the descend- ing part is increased. This we find established in the modern 248 AEC OF THE MERIDIAN MEASURED IN PERU. works on Dynamics : see Poisson's Traitt de Mfaanique, Vol. I. pages 348... 361. Bouguer treats on the diminution of attraction at different heights above the level of the sea. He finds that on a mountain at the height h above the level of the sea, the attraction is pro- g portional to (r- 27*) A + - 7;S, where r is the Earth's radius, A the Earth's mean density, and 8 the density of the mountain. This is the first appearance of the formula, which has now passed into elementary books; see Statics, Art. 219. On pages 364... 394 we have an account of the observations made by Bouguer and La Condamine to determine the attraction of the mountain Chimborazo. A deviation of about 7J" in the situation of the plumb-line seemed to be produced ; but this was much less than might have been expected. The mountain there- fore must contain great cavities, or be composed of materials of comparatively small density. It is plain, however, from the account that the observations were scarcely adequate to settle the matter ; nor does Bouguer himself appear to lay much stress on them. The work of Bouguer exhibits some tendency towards un- necessary speculative refinements, and will require careful atten- tion in order to master its complexity ; but nevertheless, both on practical and theoretical grounds, it may be justly considered the most important of all which the Peruvian expedition occa- sioned, and as that which should be selected by a student who desires to confine himself to one of the original accounts. 364. If we consider the whole transaction we shall have abundant reason to commend the patience and devotion which the history of the expedition clearly manifests. Ten years of exile from Paris, for a Frenchman and an Academician, formed a costly sacrifice to science ; and in this case the exile was aggra- vated by incessant labour, anxiety, and suffering. The result remains to this day one of the principal elements in the numeri- cal facts of the subject; and while we must be grateful to the two who mainly obtained it, we may pardon them if by contests which harassed only themselves they shewed how easy it is for human infirmity to tarnish the noblest names and the brightest deeds. CHAPTER XIII. D'ALEMBERT. 365. THE subjects of Attraction and the Figure of the Earth engaged much of the attention of D'Alembert : in the present Chapter and a subsequent Chapter we shall consider his researches in order. We begin with his Traite de Vequilibre et du mouvement des fluides. The first edition was published in 1744 ; the second in 1770 : both are in quarto. The first edition has a Preface which occupies xxxii pages, including the Title-leaf; then a Table des Titres ; then the text of 458 pages, followed by a page of Correc- tions. The second edition has an Avertissement which occupies a page, followed by a reprint of the preface to the first edition, and a Table dts Titres ; then the text of 476 pages. The text of the second edition is a reprint of that of the first, with some ad- ditions which furnish references to researches made by D'Alembert since the publication of the first edition of the work. 366. The only part of the edition of 1744 which directly concerns us is a section on pages 47... 51, entitled De requilibre des Fluides, dont la surface superieure est Courbe. D'Alembert says that this matter is important on account of its connexion with the question of the Figure of the Earth. Huygens had taken for the principle of equilibrium the perpendicularity of gravity at the surface. Newton used the principle of the equi- librium of central columns. Bouguer and Maupertuis shewed that both principles must hold for equilibrium. Clairaut had used the principle of canals ; and had also shewn that the thickness of a level film must be inversely proportional to the resultant force at the point. 250 D'ALEMBERT. It will be seen that in this brief sketch D'Alembert names Huygena before Newton : see Art. 65. 367. After his brief sketch of the history of the theory of fluid equilibrium D'Alembert says on his page 48 : Les differentes Loix d'equilibre, decouvertes par les Savans Georae- tres que nous venons de citer, paroissent etre les seules auxquelles nous devions nous arreter pour le present, jusqu'a ce que 1'Experience, ou une connoissance plus parfaite de la nature des Fluides nous ait persuade qu'il n'y en a point d'autres, ou peut-etre nous en fasse decouvrir d'autres. It will be seen from this extract that D'Alembert knew that certain conditions were necessary for fluid equilibrium, but did not know what conditions were sufficient He proceeds to offer certain conjectures which we now know to be inadmissible. He seems half inclined to believe that when fluid is in equilibrium the bounding surface must be plane or spherical, and the resultant force constant at all points of the surface. D'Alembert says that one of the best methods of deciding the question, at least in part, would be to shew that the Figure of the Earth found by theory agrees with that found by actual measure- ment. He adds on his page 51: Car on ne sauroit douter que la Terre ne soit applatie vers lea Poles, apres les operations si exactes qui ont etc faites au Nord, opera- tions confirmees par celle qu'a faite M. Cassini de Thury en 1740, et de laquelle il a conclu 1'applatissement de la Terre, sans egard pour plusieurs mesures pr6cedentes, d'ou resultoit le contraire, et qu'apparemment il n'a pas cru assez exactes. 368. It will be convenient to notice here the additional re- marks on our subject which occur in the edition of 1770 ; although we thus disturb the order of chronology. On his page 36, D'Alembert objects to Clairaut's apparent belief that the laws of Hydrostatics required the denser strata of the Earth to be the nearer to the centre ; D'Alembert refers to page 280 of Clairaut's work, and he might also have referred to other pages. See Art. 315. D'ALEMBERT. 251 The section which we have cited in Art. 366 is enlarged in the second edition. The names of Maclaurin and Daniel Bernoulli are mentioned as having in effect before Clairaut given the prin- ciple, that the fluid in any canal with its ends at the surface of the fluid must be in equilibrium. But D'Alembert allows that Clairaut was the first to develop the use of the principle. D'Alembert adds, with reference to Clairaut, on his page 50: Je crois au reste, que ce Savant s'est trompe", quand il a avance que dans un Fluide heterogene, les couches de differente densite devoient toutes etre de niveau. Voyez a ce siijet 1'art. 86 de mes Recherches sur la cause des vents, et man Essai sur la Resistance des Fluides, art. 165, 166 et 167. II est vrai que je me suis aussi trompe inoi-meme, en croyant que dans le systeine de 1'Attraction, les couches de la Terre pourroient n'etre pas de niveau. C'est ce que le cel^bre M. de la Grange a reraar- que dans le second volume des Memoires de la Socie"te Koyale des Sciences de Turin, et ce que je prouverai moi-meme ailleurs plus en detail. Mais il n'en est pas moms vrai, que dans un grand nombre d'hypotheses, un Fluide peut etre en equilibre, sans que les particules d'une meme densite se trouvent necessairement placees dans une couche de niveau. Quoi qu'il en soit, il est constant, suivant le Principe general dont on vient de parler [the principle of canals], que chaque couche de niveau doit etre egalement pressee en tous ses points ; et qu'ainsi 1'epaisseur en chaque point doit etre en raison inverse du produit de la densite par la pesanteur. See Art. 815. D'Alembert in fact admitted his error in 1768: see his Opuscules Mathdmatiques, Vol. v. page 2. I have not found where he returns to the subject after 1770, as we might expect he would from his words above, "je prouverai moi-meme..." Perhaps it really refers to what he gave in the fifth volume* of the Opuscules Mathematiques, and was written before, though published after, that volume. There is another memoir on Fluids in the eighth volume of the Opuscules Mathematiques, but it does not seem to bear on this point. 369. I will notice some matters of interest which have pre- sented themselves in reading the Articles 1...58 of D'Alembert's Trait^... des Fluides. In his Article 2 he criticises, and I think justly, a demon^ 252 D'ALEMBERT. stration given by Newton, namely, the second case of Proposition 19 in the second Book of the Principia. In his Article 13 there are some remarks to shew the insuf- ficiency of two common demonstrations of the proposition that the resultant force at any point of the surface of a fluid in equilibrium Inust be perpendicular to the surface at that point. The first demonstration stands thus : if the force be not per- pendicular the tangential component will tend to move the particle on which it acts, and the fluid will, as it were, descend an inclined plane. D'Alembert objects that a set of equal balls might be placed, one above the other, and be in equilibrium on an inclined plane ; so that if a fluid be composed of such particles it would appear that the fluid might be in equilibrium with its upper surface inclined to the horizon instead of being horizontal. The second demonstration rests on the assumption that for equilibrium the centre of gravity should be as low as possible. D'Alembert brings forward two exceptions; in one the centre of gravity is at a maximum height, and in the other some forces act besides gravity. Thus in fact D'Alembert's objections hold against the improper extension of a certain theorem, and not against the proper enunciation of the theorem. See Statics, Chapter XIV. A remark made by D'Alembert in his Article 18 deserves, I think, the attention of modern elementary writers. Suppose we have a conical vessel and a cylindrical vessel with equal bases ; let them be filled with water to the same height : then the pres- sures on the bases will be equal. A popular mode of establishing this proposition amounts to taking the cylindrical vessel with its water, and then supposing a certain part to become solid, so as to leave a conical interior of fluid. D'Alembert says in substance that we ought not to assume that the pressure is unaltered by this solidification of part of the fluid : for suppose we solidify a complete horizontal lamina of the fluid, we can thus in effect remove from the base the pressure of all the fluid above this lamina. I observe some modern writers adopt the reverse order ; they begin with the conical vessel and afterwards dissolve the sides, D'ALEMBERT. 253 instead of beginning with the cylindrical vessel and solidifying : but it may be fairly doubted if the process is more satisfactory in this way. D'Alembert's Article 26 calls for some observations. We will give an account of his investigation in modern language. Let a mass of fluid be acted on by a force the direction of which is constant, but not necessarily the intensity. Take the axis of x parallel to this fixed direction ; let X denote the force at the distance x from the origin, p the pressure there, and p the density. We have then, as is well known, therefore p = IpXdx = ^r (x) say. Suppose the fluid to be enclosed in a vessel of any shape, the ends being plane figures at right angles to the axis of x. Take i/r (x) such that it vanishes at one end. If ty (x} is such that it vanishes also at the other end, and is never negative, the ends may be removed without destroying the equilibrium : this is obvious. But if ty(x) can become negative, equilibrium will not hold when the ends are removed : this is also obvious. Suppose, then the ends to remain. D'Alembert says that the pressure at the end for which ^ (x) vanishes will be numerically equal to the greatest negative value of i|r (x). This is inaccurate. The pressure cannot indeed be less than this, but may be as much greater as we please. In fact we may take p= C + ty(x), where C is an arbitrary constant: and provided C be large enough to ensure that p is always positive, equilibrium will subsist. The value of the pressure at the other end will then be deter- mined by ascribing the proper value to x in the expression C + -ty(x): but D'Alembert seems to say that the pressure will be ty (x). 370. The next work by D'Alembert which we have to exa- mine is his Reflexions sur la Cause ge'ne'rate des Vents. This work 254 D'ALEMBEKT. was published in 1747; it gained the prize proposed by the Berlin Academy for 1746. The work is in quarto. There is a Title-page, a Dedication, and an Avert issement ; an Introduction of xxviii pages ; then 194 pages which contain a French transla- tion of the original essay with some additions ; and lastly, 138 pages which contain!, the original essay in Latin. In our remarks we shall confine ourselves to the French translation. 371. The dedication is to Frederic, called the Great ; and is in the usual adulatory strain of these objectionable compositions. The introduction gives a general account of the\ contents of the essay, intended for the use of readers with little mathematical knowledge. Two sentences are of sufficient interest to be re- produced. One sentence offers a curious reason for referring the winds to the action of the Sun and the Moon ; it occurs on page ii. After stating that the ebb and flow of the tide are admitted to be due to this action, D'Alembert says : Quel que soit le principe de cette action, il est incontestable que pour se transmettre jusqu'a 1'Ocean, elle doit traverser auparavant la masse d'air dont il est environne, et que par consequent elle doit mou- voir les parties qui coinposent cette masse. The other sentence relates to the difficulty which the Cartesians found in admitting that the attraction of the Sun or of the Moon could produce high water simultaneously on the meridian under the attracting body, and on the opposite meridian. D'Alembert says, with zeal amounting to anger, on his page x : La preuve simple et facile que je viens de donner du contraire, sang figure et sans calcul, aneantira peut-etre enfin pour toujours une objection aussi frivole, qui est pourtant une des principal es de cette Secte centre la The'orie de la gravitation universelle. 372. In the work itself we first notice pages 11... 17. These contain an approximate solution of what we may call a companion to Huygens's problem. D'Alembert enunciates it in the most general form, namely, where the attractive force is any function of the distance from a fixed point; but in his solution he finds it D'ALEMBERT. 255 sufficient to take the force constant. See Arts. 55, 56, and 173. Let o> denote the angular velocity, f the constant central force, c the radius of the sphere which the fluid \vould form if there were no rotation ; then assuming that o> 2 e is small compared with f t the surface will be a spheroid, and the equation to the gener- ating curve will be , V where r is the radius vector from the centre of force, and 6 is the angle which r makes with the axis of revolution. This result may be easily deduced from that given in Art. 55. D'Alembert him- self solves the problem by what we should now call a method of Virtual Velocities. D'Alembert finds the volume of the solid bounded by the sphe- roid, the sphere of radius c, and the double cone having its vertex at the common centre, and having the semi- vertical angle 6 : see ,. !- rm i, A- 27ro)V cos sin 2 ,. . his page lo. The result in our notation is - - ^ - ; this V may be easily verified. In this expression some of the volume is estimated negative if 6 be so great that we get beyond the value for which the sphere and the spheroid intersect. 373. We have no concern with the discussions on the motion of a fluid, to which D'Alembert now proceeds, so that we pass on to pages 33... 45 of his work. D'Alembert determines the form of relative equilibrium of a thin layer of fluid spread over a solid spherical mass ; taking the action of the fluid itself into account, and supposing uniform rotation. D'Alembert requires the attraction of a homogeneous oblatum, which is nearly spherical, on a particle situated at any point of its surface. This he obtains by three steps. (1) He quotes a theorem given by Maclaurin in his Essay on the Tides, by which the attraction on a particle at any point is known, when it is known for a particle at the pole and for a particle at the equator. See Art. 244. 256 D'ALEMBERT. (2) He has an approximate investigation for finding the at- traction on a particle at the pole. This was originally given by Clairaut, but D'Alembert does not refer to him. See Art. 233. (3) He has an approximate investigation for finding the at- traction on a particle at the equator. He mentions Daniel Bernoulli in connexion with this ; but the principle is the same as in the investigation for the particle at the pole, first given by Clairaut. 374. We will now furnish in modern language, and in our own notation, an equivalent to D'Alembert's process. Suppose s the radius, and cr the density of the central sphere, and p the density of the fluid. We may consider that there is an oblatum of density p, and also a sphere of density a p. Let the ellipticity of the oblatum be e, which is supposed small ; let x and z be the coordinates of a point parallel respect- ively to the major and minor axes of the generating ellipse ; then the attractions of the oblatum in these directions will be, by Art 261, respectively Put the first in the form ~- 1 1 -f ) r, -- - x. Then on 3 \ o / o the whole we have a force towards the centre, the value of which is the product of the distance into 5- f I + J ; together with the force ~- x parallel to the major axis outwards from the minor axis. Thus we see that we can avail ourselves of the solution of the companion to Huygens's problem, provided we add - to the o> 2 , and use the proper value of the central force. This central force 4?rp /, 4e\ 4?r . N s 3 at the distance r will be -3*- ( 1 -f y ) r + ^ (cr p) -^ . D'ALEMBERT. 257 Hence, as by Art. 55 we have e = | , we now obtain therefore e = 4e For an approximation we reject in comparison with unity in the denominator ; and indeed our investigation is not accurate enough to justify us in retaining this term : thus c = STT . to . V r ^ ^' ? D'Alembert's own process is ruder and he has - instead of our -3 in our notation. As yet we have not introduced the condition that the layer of fluid is thin ; suppose it so thin that s may be taken equal to r in the denominator : thus = (l-*P\ I-*?' ( W 5o- where rj is what would be the ellipticity if the attraction of the fluid itself were entirely neglected. 375. On his page 40 D'Alembert proceeds to some remarks on the Figure of the Earth ; for these he had prepared us on his page 10, saying, "...ou je de*montre plusieurs ve'rite's fort para- doxes sur cette matiere." The remarks amount in substance to T. M. A. 17 258 the two obvious statements that the value just found for e is very large if 3/3 is nearly equal to 5<r, and will be negative if 3/> is greater than 5(7. If e is not numerically small, our approxima- tions do not hold. If e is negative and numerically small our supposed oblatum is really an oblongum. D'Alembert seems to consider it rather singular that an ob- longum should be a possible form for the surface. See his page 41. 376. D'Alembert next considers the case in which the nucleus is not a sphere but an oblatum ; the process is less satisfactory than that in Art. 374, because we have now to deal with the attraction of an oblatum on an external particle. Suppose, how- ever, that the layer of fluid is very thin ; let the ellipticity of the solid oblatum be small, and denote it by e'. Then we see that we shall obtain an approximation to the required result by adding -=- ((7 p) e to o> 2 ; so that 5 w 2 4?r 3 M 5<7/ 377. The result just obtained is one to which D'Alembert seems to have attached great importance. It must be observed, however, that it is only a particular case of a general formula given by Clairaut. Take the final result of Art. 323 : in the inte- grals represented by A and D let the density be constant, and denote it by cr. Thus therefore, ) = ( - ft ) V+ Sj (en-" this is in fact given in Case II. of Art. 324. We have here then the more accurate form : if we now suppose that the difference between r and r t may be neglected, we obtain e, (10<7 - 6 ft ) = 6e' (a- - ft) + 5j<r, 259. which agrees with D'Alembert's result ; it is more simple but less accurate than the immediately preceding form. D'Alembert him- self subsequently obtained the more accurate form: see his Recherches...Sy steme du Monde, Vol. III. page 225. Clairaut was content with somewhat less than he might have deduced from his own formula ; see Art. 328. 378. The value of e obtained in Art. 376 may be negative ; it will be negative if the numerator is positive and - - is greater than unity. D'Alembert says on his page 42, ... Done si la Terre etoit un Spheroide allonge, il ne seroit pas absolu- ment necessaire d'avoir recours pour expliquer ce Phenomena, a un noyau interieur allonge. Car il pourroit se faire que ce noyau fut applati, et que la Terre fut allougee vers les Poles. This remark is probably aimed at Clairaut ; see Boscovich De Litter aria Expeditione... page 464 : we have, however, shewn in Art. 326, that Clairaut might have drawn the same inference if he pleased. But Clairaut had a conviction of the propriety of assuming the Earth to be densest at the centre ; and thus he would naturally neglect any hypothesis which; was inconsistent with this conviction. With respect to the formula of Art. 376, D'Alembert remarks that if 50- Bp = 0, and also 17 + - (1 - J e' = 0, then 6 may have any value we please, provided only it be small : he repeats this remark in his Recherches . . . Systeme du Monde, Vol. ill., page 190. 379. D'Alembert makes a statement at the top of his page 44 which I do not verify. He proposes to estimate the force on the fluid in the direction of a tangent at any point of the meridian of the nucleus. Let / denote the force to the centre, 6 the angle between the axis and the radius vector to the point, then the re- quired force is the product of sin 6 cos 6 into j. c+ -V-^eh-2/e, 172 263 D'ALEMBERT. that is 2/(e-e') sin 6 cos 0, that is 2/sin 6 cos that is - 2 sin cos 6. 3 f 5(7 2/e' D'Alembert omits the term -~ . In fact the force along the o tangent must vanish if e = e ; but D'Alembert's expression would never allow it to vanish. 380. We proceed to pages 151... 158 of the Reflexions sur... Vents, which contain some new and interesting matter relating to attractions. D'Alembert obtains, in effect, formulae for deter- mining the attraction at any point of the surface of an ellipsoid which is nearly spherical. He first states what the results are for points at the ends of the three axes; he does not give his investigation, which was probably of the kind which he attri- buted to Daniel Bernoulli : see Art. 373. Let the three semi- axes be r, r /3, r 7, where /3 and 7 are small : it is easy to shew by this method that the attraction at the end of the first axis is -r^(/3-l-7). I f r greater symmetry, we denote o lo the semi-axes by r a, r {3, r y, where a, /3, 7 are small, the attraction at the end of the first axis is (r a) (0 a +7 a), o -Lo that is -Q- tr - ? J . In order to express the attraction at any point of the surface, D'Alembert uses, in effect, the pro- perty that the attraction perpendicular to a principal plane of the ellipsoid varies as the distance from that plane. This, he says, follows from the principles given in Maclaurin's Essay on the Tides. Maclaurin himself did not explicitly go beyond the case of ellipsoids of revolution ; but D'Alembert's extension was very obvious. D'ALEMBERT. 261 Let x, y, z be the coordinates of any point on the surface of the ellipsoid referred to the axes as axes of coordinates; let X, Y, Z be the attractions parallel to these axes : then ^r x 4vr / a + 2/3 + 2 7 \ 47ne / 4a 2/3 27\ X= r-al r - -T--J =: ^-( 1 + - or ~>' and similar expressions hold for Y and / 381. Then D'Alembert shews that an ellipsoid of homoge- neous fluid, differing very little from a sphere, cannot be in equilibrium under its own attraction ; in fact, the resultant force will not be at right angles to the free surface. D'Alembert's demonstration is laborious, but sound, if we use the correction of a mistake furnished by himself in his Opuscules Mathematiques, Vol. I. page 252. The modern method would be to form the condition which involves the direction cosines of the resultant force and of the normal to the surface. This condition is z +Z?^"' r +7F^"^*F^ J that is, approximately, x r y This condition is not fulfilled. D'Alembert some years later supposed that he had demon- strated the relative equilibrium of a rotating ellipsoid of fluid to be impossible; see his Recherches . . . System e du Monde, Vol. in. page 256 : but he forgot that the so-called centrifugal force must also be Considered. We know now by Jacobi's Theorem that such relative equilibrium is possible. Further, D'Alembert's demonstration shews that a fluid ellip- soid which is nearly spherical cannot be in equilibrium under its own attraction ; but it does not shew that this result holds for every ellipsoid. This is however the case ; for in the demon- stration of Jacobi's Theorem we shall find that the angular ve- locity has a value which cannot vanish. 262 D'ALEMBERT. 382. On his page 156, D'Alembert proceeds to the case in which a solid homogeneous ellipsoid is surrounded by a thin stratum of fluid ofSifferent density in equilibrium. The mistake already referred to influences this investigation ; and moreover D'Alembert misinterprets his results, and infers that if the solid part is a solid of revolution it must be a sphere, and that the 3 density of the solid part must be exactly ^ of the density of the o fluid. This contradicts his own investigation in pages 40... 44 of the work : see Art. 375. However, in his Opuscules Mathemati- ques, Vol. i. pages 253... 255, he corrects his errors, and is more successful. Let a- be the density of the solid, p the density of the fluid ; let 6j and e 2 be the ellipticities of the two principal sections of the solid, fj and f 2 the corresponding ellipticities of the two sections of the external fluid surface. D'Alembert obtains an approximate result which we may thus express So far he is correct, but he adds that the solid figure and the external figure are semblables, which is not admissible : to make the figures like we should require ~ and ^ both to be equal to i & unity. 383. It will be instructive to notice the principle involved in D'Alembert's treatment of this problem : I will give it in substance though not in his form. I use as before r a, r /3, r y for the semi-axes of the external figure ; and r a, r ff, r y for those of the solid part. We may then consider that we have a body with the former semi- axes, of the density p, and also a body with the latter semi-axes of the density <r p. For the former body we may take as before 4a- 2/3-27^ +- ---> 3 and similar expressions for Y and Z. D'ALEMBERT. 263 For the latter body we take *TT(<T-P)X f 4a'-2ff-2 7 '\ 3 \C' or )' and two similar expressions. This amounts to supposing the second body enlarged in size until it just passes through the attracted point ; that is in fact we introduce a thin ellipsoidal shell of density a- p. But no sensible error is thus produced ; for the action of this shell is in amount only of the first order ; and is in direction, as we now know, accurately along the normal to its outer surface. Hence the shell would supply a force along the tangent plane to the fluid surface which would be only of the second order ; and so for our purpose rnay be neglected. D'Alembert leaves his readers to think this point out for them- selves, but in a later work he supplied a hint : see his Opuscules Mathematiques, Vol. VI. page 226. Thus we take for the whole attraction parallel to the axis of x Call this Xj and let F x and Z l have similar meanings. We know that for equilibrium we must have r This leads by easy reduction to a $ _ a 7 D'Alembert then shews that if the whole mass revolve round one of the axes with uniform angular velocity relative equilibrium may subsist. Take the axis of x as that of revolution ; let &> be the angular velocity: then we must put o> 2 y to what we called Y v and o> 2 3 to what we called Z This will be found to lead to and 264 D'ALEMBERT. 384. The next work by D'Alembert which we have to ex- amine is his Recherches sur la Precession des Equinoxes... This work was published in 1749 ; it is in quarto. The Title, Dedication and Introduction occupy xxxviii pages; then follows a table of Contents, and then the text of 184 pages. There is a German translation of this work in octavo, by Dr G. K. Seuffert, published at Nurnberg, 1857. 385. "We are concerned only with Chapter IX. of the work, which is entitled Consequences qui resultent de la Theorie prfce- dente par rapport a la figure de la Terre ; this occupies pages 95. ..105. By comparing his theory of Precession with observation, P'Alembert obtained the following numerical relation dr j 1 dr 1 dr 5 , 324* p-j- dr > dr The notation will be understood from what has been said before : see Art. 323. This very important result remains almost unchanged in the modern theory ; the fraction ^-r being replaced by "00326, which differs little from it : see Rdsal, Traite EUmentaire de Me'canique Celeste, page 226. 386. D'Alembert combines his own result with one given by Clairaut on his page 226 : it is that which occurs in our Art. 327 ; denoting r t by unity, we may write it thus : f 1 dr *j at 1 d ( r *3 j K-f 1 dr *j 10e, p-j-dr 6 I p -\ ' dr =5j p-j-dr .......... (1). 'Jo dr Jo dr J J Q dr Now D'Alembert, relying on the measures in Lapland and Peru, takes e, = -=-. ; and so the result in Art. 385 may be written thus : d(r*e} , 174 f 1 dr -^ L dr = e l -^-:\ p j-dr ........ , .......... (2). dr ' r D'ALEMBERT. 265 dr* j 7 ( l dr * j /QN Assume p-j-ar = fc p-j-ar (9). Then from (1), (2), and (3) we obtain e t = 5 Now we shall shew presently that Jc is less than ~ ; so that e x 324 5 3 5 5? 289 is less than J \, , that is less than ' . . This he says Li-kV -_ 1 /4<< ' ~ makes e, less than j which is inconsistent with the value 6l = t given by observation. Instead of 256 we might put 267. Thus D'Alembert infers that the Earth cannot be composed of solid elliptic strata, which is the hypothesis on which the result quoted from Clairaut was obtained. We know now that e l cannot be so great as = ; and thus the contradiction which D'Alembert points out no longer exists. 387. We shall now shew, as we have stated, that Ic is less than . We have to shew that o ri d r 5 ri fo* 31 p j- dr is less than 51 p - 7 - dr, J o "* Jo r dr where the symbols denote positive quantities. D'Alembert spreads the demonstration over six pages. He makes three cases; that in which p always decreases as r increases from to 1, that in which p always increases, and that in which p sometimes decreases and sometimes increases. But the required result can be obtained instantaneously. We have to shew that ri ri I pr*dr is less than I pr^dr, Jo Jo or that I pr* (r 2 - 1) dr is negative ; and this is obvious, for every element of the last integral is negative. 266 D'ALEMBERT. 388. We may also shew that if p always decreases as r in- creases from to 1, then r \ J i ftr* p - 7 - dr is less than I p - dr. r dr Integrate by parts: let p x be the value of p at the surface. Then we have to shew that Pi - r ;r r * dr is less than A - P ^ f*dr, Jo dr Hl J dr or that I -^ ?- 3 (1 - r 2 ) dr is negative ; r and this is obvious, for -~ is negative by supposition, so that every element of the last integral is negative. 389. D'Alembert's page 101 is not intelligible to me. I imagine he means to say that perhaps some person will be able to ri dr* shew that if p increases constantly from the centre I p -y- dr is less than (o~^j I P~J~dr, where /3 is some positive quantity. This we have shewn in Art. 388, where ^ yS is equal to unity, o 2 so that ft = Q . o 390. D'Alembert then considers on his pages 103... 105, whether the facts and the theory will agree on the supposition that the Earth consists of a solid elliptic mass covered with a thin layer of fluid. We must observe that the layer here is to be of finite thickness though thin ; the case of an infinitesimal layer was in fact that which was dismissed as untenable in Art. 386. D'Alembert assumes without any adequate investigation that the action of the fluid on the solid will not affect the Precession. See on this point Re'sal, Traiti Elementaire de Mtcanique Celeste, pages 353... 356. DALEMBERT. 267 As in Art, 376, we have here e is the ellipticity of the exterior surface of the fluid, and e' the ellipticity of the solid nucleus. Thus 3 therefore 174 If we take e' less than ^^ we find - to be positive ; the num- (7 her ^^ is that which presented itself in Art. 386 ; but it appears to me quite arbitrary to introduce it here. D'Alembert, however, has no misgiving: see his page 105. 391. D'Alembert gives the following inequality on his page 99 : If a? is a proper fraction, 2 is greater than a? (5 3# 2 ). He establishes it easily by taking the differential coefficient of x 5 (5 - 3aty We can establish it by common Algebra. For 2 - x * (5 - 3# 2 ) = 2 (1 - x 5 ) - 5x 3 (1 -a 2 ) = (1 -a?) {2 (1 + x + x*+ a? + x 4 ) - 5x 3 (1 + x)} this is necessarily positive. The last expression may be put also as (1 - xf {2 (1 + x) (1 + x+ x*) +x* (2 + x)}, that is as (1 - x) 2 {2 + 4>x + 6# 2 + 392. The next work by D'Alembert which we have to ex- amine is his Essai d'une Nouvelle Theorie de la assistance des Fluides. 268 D'ALEMBERT. This work was published in 1752 ; it is in quarto. The Title, Dedication, Introduction, and Title of Contents occupy xlvi pages ; the text occupies 212 pages. The work was composed in competition for a prize proposed by the Academy of Berlin. The Academy instead of awarding the prize requested the candidates to give supplements shewing the agreement of their theories with experiments. D'Alembert seems to have been not quite satisfied with this proceeding ; he resolved to abstain from a new competition, and to publish his essay at once. He adds, on his page xl : Je souhaite par 1'interet que je prends a Favancement des Sciences, que les Juges nommes par cette illustre Compagnie, et qui n'ont pas sans doute propose cette question sans s'assurer si la solution en etoit possible, trouvent pleinement de quoi se satisfaire dans les Ouvrages qui leur seront envoyes pour le concours. 393. The second Chapter of the book is entitled Principes gdneraux de Vdquilibre des Fluides ; it occupies pages 13... 18. D'Alembert first adverts to the principle of Canals; he de- duces Clairaut's condition with respect to curved canals from Maclaurin's with respect to straight canals. To a modern reader the principle seems sufficiently evident without any remark. 394. D'Alembert establishes an important result which cari be best explained by the aid of the modern equations for fluid equi- librium. Confining ourselves for simplicity to the case of forces in one plane we have dp v dp -f- = pX y -f dx dy from these it follows that D'Alembert demonstrates this condition ; for the particular case in .which p is constant it was already known, as we have .seen in Art. 306. D'Alembert considers his own demonstration simpler than any which had yet been given. DALEMBERT. 269 D'Alembert himself does not use the symbol p or speak of the pressure of the fluid. It will however be interesting and instruc- tive to give the essence of his investigation in modern language. Let the coordinates of any point P be a; and y ; let the coor- dinates of an adjacent point R be x -f h and y + k. Complete the rectangle PQRS, having its sides parallel to the axes. Let p be the density at P, let p t be the mean density along PQ, and p 2 the mean density along PS. Let p be the pressure at P; then the pressure at Q will ulti- mately be p 4- pi Yk, and the pressure at S will ultimately be p + p z Xh. Now we may form two expressions for the pressure at E, one obtained by passing from Q to R, and the other obtained by passing from S to R. The former expression is ultimately and the latter is equate these and we obtain ultimately dx that is This mode of giving as it were a physical interpretation to the condition just obtained might be called D'Alembert's hydrostatical 270 D'ALEMBERT. principle ; though it is not very clearly put by himself. We may say verbally that the principle amounts to this : the change of pressure in passing from one given point of a fluid in equilibrium to another is independent of the path by which we proceed. 395. An Appendix entitled E^flexions sur les loix de bre des Fluides occupies pages 190...212 of the work. D'Alembert gives on his pages 190... 194 another demonstra- tion of the equation -j- (p Y) = -j- (pX) ; this demonstration is sound but complex: he gives it, he says, because it will supply the oppor- tunity for some important remarks on the laws of the equilibrium of fluids. The remarks do not seem to me of great importance; but the reader can judge for himself from the account which will now be given of them. 396. D'Alembert says on his page 195, in effect, that if with previous writers on this subject we suppose the density to be con- stant throughout eveiy level surface we arrive at the equation -y- =-j- instead of that in Art. 394 : this appears to him to re- quire explanation. Along a surface of equal density we have -~ dx + -J- dy = ; if this surface is also a level surface we have Xdx + Ydy = ; hence F^ = X & , and the equation of Art. 394 ax ay reduces to -j- = -=- . So far he is right, but he adds a remark dx dy which is quite erroneous ; changing his notation to that which we have used, his words are : - M f dX dY Mais il taut remarquer que lequation - = -=- n a lieu dans ce cas que pour les couches ...... auxquelles la direction de la pesanteur est per- pendiculaire, au lieu que 1'equation -=- (pY) = y (pX) a lieu gnrale- ment pour telle couche qu'on voudra... This is a strange error: from the nature of the equation /7V fJJC -T- = -p it is quite independent of direction. D'ALEMBERT. 271 397. D'Alembert says on his page 197, that the equation of Art. 394 supposes p, X, and Y to be functions of x and y : but he does not see why we should be restricted to this hypothesis. He proceeds to something which he considers more general, but which is really not so ; in fact he supposes that X and Y are functions of x, y, and f, where f is itself a definite function of x and y : but it is obvious that this is practically identical with the usual hypothe- sis. I found after I had written this that Lagrange had made an equivalent remark in the Miscellanea Taurinensia, Vol. II. page 282. D'Alembert himself also "subsequently admitted that this intro- duction of was superfluous : see his Opuscules Mathematiques, Vol. vin. page 16. 398. D'Alembert makes an erroneous statement on his page 199, namely, that if the pressure be equal at all points of the bounding surface the force must be equal at all points : we know that this is not necessarily the case. Indeed D'Alembert himself says on his page 201 : ... A 1'egard du principe de 1'egalite des forces, il est evident que s'il e"toit admis, toutes les Theories qu'on a donnees de la Figure de la Terre, en la considerant comme un FluicTe, et en ayant e"gard a 1'attraction des parties, et a la rotation de 1' Axe, devroient etre regardees comme fausses. 399. D'Alembert returns to the matter which we noticed in Art. 367 ; and seems still half persuaded of the truth of the absurd opinion stated there. However he converts himself from his error by the aid of an important principle which he had formerly given. The following is the substance of his argument : it is obvious that a fluid may be in motion without having its surface plane or spherical; and it follows from what we now call D'Alembert's Principle that if any motion is known we know also the forces which would maintain the system in equilibrium in the configura- tion which it has at any instant ; thus forces do exist which would maintain a fluid in equilibrium and give to the surface a form which is neither plane nor spherical. 400. D'Alembert seems to attach great importance to the fact that if a fluid be in equilibrium the surfaces of equal density 272 D'ALEMBERT. are not necessarily level surfaces. We know now, with the usual notation, that if Xdx + Ydy + Zdz is a perfect differential, the surfaces of equal density will be level surfaces ; moreover for such forces as occur in nature this condition is satisfied : hence for such cases as occur in nature it is true that the surfaces of equal density are level surfaces. But D'Alemhert's statement is correct, that surfaces of equal density are not necessarily level surfaces. See Arts. 315 and 368. 401. We will give briefly the example which D'Alembert discusses, translating his process into modern language. Suppose s the distance of a point from the origin, and 6 the angle which s makes with a fixed straight line. Let S denote the force along s, and T that at right angles to s ; and let <r denote the density. Then the usual equations for the equilibrium of a fluid are , ds sdd where p denotes the pressure. Therefore This condition in fact agrees with what D'Alembert himself deduces from the principle of canals. Now let us assume that the fluid is arranged in strata of equal density; let the curve of equal density be determined by the equation 8 = r+apZ. ............................. (2), where r is a parameter which particularises the curve we consider, p is a function of r, and Z a function of ; and a is a very small quantity, the square- of which will be neglected. Also suppose that -8=p' + ap"Z', and T=ap'"Z" ............... (3), where p', p", and p" are functions of r\ and Z' and Z" are functions of 6. The notation is kept very close to D'Alembert's, though not exactly the same. D'ALEMBERT. 273 Now (1) may be written cfcr dS ^ d<r dT The condition that a- is constant along the curves determined by (2) gives d& da- ds __ dO ds dd ,, , . da dZ da- that is, ~jA~^~ a P ~jn ~j~ ~ " du do ds Then (4) becomes ^ da- dS Substitute from (3), and neglect the square of a ; thus ,dZ , nr . n .d(T Here -^ means the differential coefficient of S with respect to 0, supposing s constant ; and so it is found by combining dS (dp , , dp"\ dr , dZ' ds / ' rd\ dr dZ and ^ Hence, neglecting the square of a, dS do' dZ dZ' -dd = - p Tr W +ap ~Te> also if we neglect the square of a we may put -y- for in (5). Then, dividing by a, we obtain ,dZ ,,,~,,.cZ<r , dp dZ bp-30~ T P Z ^dr = ap Z -"^M dZ' r,,,dp" x _, - ............ (6). T. M. A. 18 274 D'ALEMBERT, 402. We will make some remarks on the equation (6). D'Alembert himself by transposition puts it in this form : dZ d f , x ~,, d . ,, dZ' , dp dZ D'Alembert obtains this result by the method which we have exemplified in Art. 394. In modern language we may say that he passes from one point of the fluid to another by two different routes; and thus he obtains two expressions for the change of pressure, which can be equated. But as he does not use the word pressure, or the symbol p, his method is somewhat obscure. In the diagram of Art. 394, we see that the increase of pressure from P to Q -f increase from Q to E = increase from P to S + increase from S to R. With D'Alembert the equivalent statement takes the less natural form, the increase of pressure from Q to R - increase from P to S = increase from S to E increase from P to Q. Instead of the words increase of pressure from P to Q, D'Alembert uses such words as force of the column P Q along PQ ; and these seem scarcely intelligible. D'Alembert attempts to enunciate this case of his hydrostatical principle in words in his Recherches...Systeme du Monde, Yol. III. page 226, where he says : ...ilfaut supposer la difference de pesanteur de deux couches de niveau infiniment proches, e"gale a la difference de pesanteur de deux couchea verticales infiniment proches,... An enunciation, partly in words and partly by symbols, is also given by Lagrange ; see the Miscellanea Taurinensia, Vol. n. page 285. We may remark that D'Alembert's notation might be rendered at once simpler and more general. Instead of pZ, where p is a function of r and Z a function of 6, put F, where F is a function of both r and 6 ; also put V instead of p"Z', and V" instead of p"Z". Then the equation at the beginning of this Article may be written dV d . , d , ,,, dV D'ALEMBERT. 275 In his Opuscules MatMmatiques, Vol. V. page 6, D'Alembert returns to the example of Art. 401. There he takes p to be a function of s instead of r ; or, which comes to the same thing to his order of approximation, he puts instead of the first of equa- tions (3) j f j r7 hence we have an additional term a -r- p -JTT a- on the right-hand ar au side of (5) : and finally, instead of (6), we obtain ,dZd<r d , ,,dZ' 403. I am not sure that I understand D'Alembert' s conti- nuation after the point which we reached at the end of Art. 401 ; but I think that it is substantially equivalent to the following. Assume that the surfaces of equal density are level surfaces ; then the force along the tangent to the curve considered must vanish. Thus we obtain to our order of approximation pp dZ , _ r M-P z Now p and p are functions of r only, and Z and Z' are func- tions of 6 only ; so we must have fit _ fV" 7" r ^ (*7\ - Cp , Z -C m ..................... (7), where C is some constant. Substituting in (6) we obtain ,,dZ' which, as before, leads to dZ' -rdZ where B is some constant. Thus we have the four equations (7) and (8) holding in place of the single equation (4). 182 276 D'ALEMBERT. From the first of (8) we have Z' = BZ+ B', where B' is some constant. From the first of (7) and the second of (8) we get d(pp) d P _ -n dr~~ p dr~ ** so that -p"=p'&. Thus, finally, that is, where B^ is some constant. These results are of course less general than the single equa- tion (4). 404. D'Alembert finishes the Appendix with some matter which is very closely connected with our subject. He says on his pages 208 and 209 : Je remarquerai a cette occasion, qii'il me semble qu'on n'a point encore resolu d'une raaiiiere assez gcnerale le Probleme de la figure de la Terre, dans 1'hypothese que 1' attraction soit en raison inverse du qnarre des distances, et que la Terre soit composee d'un amas de Fluides de differentes density's. Accordingly, D'Alembert proposes his more general solution of the problem of the Figure of the Earth. It would not be ad- visable to devote much space to shew that D'Alembert's additions to Clairaut's investigations are worthless ; but as we have already given the principal formulae which are necessary, we shall be able with brevity to justify this opinion. D'Alembert himself refers, as we shall do, to Clairaut, for some formulae which are necessary. D'ALEMBERT. 277 We adopt Clairaut's hypothesis that the Earth consists of ellipsoidal fluid strata of varying density and ellipticity. Let p denote the density; take the known equation of Art. 401, CD- Here the quantities are supposed to be expressed in terms of and s ; and we use the square brackets to indicate this. But suppose that s is changed into r (1 + e sin 2 6) ; then we have to transform (1) suitably. and -JQ is to be found on the supposition that s is constant, so that to our order of approximation it is equal to 2re sin cos 0. Thus rejecting the square of e we have from (1) ^ (p8) - | (pS) 2re sin 9 cos = (prT). Let <f> denote the angle between the radius vector and the tangent to the ellipse at the point considered ; so that to our order cos <f> = 2e sin 6 cos 6. Hence - 2 sin Thus ~ ( P S) + 2/3 sin Q cos 68 ^ = - (pr Q), d\j U'f CUT where Q stands for T+ cos $8, that is, for the whole force along the ta Hence finally along the tangent ; and the first term may be written p -j^ . do (2). This equation substantially coincides with that which D'Alembert uses ; but he does not sufficiently explain his process. 278 D'ALEMBEKT. 405. We have now to give the values of Q and 8. I shall use the following notation : TO) for [ r pr*dr, Jo Let b be the extreme value of r, that is the value of r at the surface ; and let o> be the angular velocity. Then it will be found that M3 sin 2 6- 2) The value of Q is found as in Art. 341, or Clairaut's page 273. The value of -S is found as in Art. 336, or Clairaut's page 247: it is only necessary to add to what is there given the central attraction which arises from the matter which may be said to be external to the attracted point, and thus we obtain the term which involves H in the manner the term involving O 5 was obtained. dS Hence - Iftrr Thus (2) becomes to our order of approximation D'ALEMBERT. 279 Let K = ~ T(r) -^k^- - ^ [fl (ft)-fl (r)] - ^ 5 then multiply 7* O7* oTJ" by r 4 and differentiate ; then divide by r* and differentiate again. Thus we obtain d*e 2pr* de [6 2pr \ r* d (I &_fj*jt\ m Moreover (3) may be written multiply by - and differentiate : then we obtain ok __ f 6 _ v Comparing (4) and (5) we obtain -j- (- - -\ =0; therefore - - = M & constant ; and so the right-hand side of (4) becomes p ar r* d f 1 d (Mp dr < dr(r dp Thus D'Alembert considers he has found a more general result than had hitherto been given ; for we know that Clairaut's de- rived equation agrees with (4) when the right-hand side is changed to zero : see Art. 343. But D'Alembert himself admits, that at the external surface there can be no tangential force, and so K must vanish there ; see the last line of his page 211. This would suggest M= 0; but D'Alembert wishes to avoid this, and so he says it will be sufficient to have -:- infinite at the external surface. dr The error involved is very serious even for D'Alembert: such a strange result should have led him to review his process. If we develope the right-hand side of (3) we have one term involving 280 D'ALEMBERT. ~- , and another involving p ; the latter term is exactly the same U/T as we have on the left-hand side of (3). Thus (3) becomes simply, in D'Alembert's notation, Kr -~ ; thus either K= 0, or -/- = ; ar ar in the latter case the density is constant : in both cases the level surfaces are surfaces of equal density. In fact, as we stated in Art. 400, we know that for such forces as occur in nature the level surfaces must be surfaces of equal density; this was pointed out by Lagrange in some observations on D'Alembert's misconception : see the Miscellanea Taurinensia, Vol. II. page 285. 406. D'Alembert himself briefly admitted and corrected his error in his Opuscules, Vol. V. page 4 : my remarks were written before I had arrived at this admission; and I have ventured to retain them. It is curious to notice the complacent satisfaction with which D'Alembert, up to the period of the admission of his error, regarded his efforts to improve the important result which I call Clairaut's derived equation: see the Recherclies....Systeme du Monde, Vol. II. page 290, and Vol. III. pages xxxvi and xxxvii ; and also the article Figure de la Terre in the original Encyclope'die. 407. We might have deduced equation (3) of Art. 405 from equation (6) of Art. 401. Return to the notation of Art. 401, using o- for the density. We have and thus equation (6) becomes d , , , /, , , dp fo<r(pp -rp ) = ap + trp ^ ; and a/) = re, p = 4-rr \ ~ + small terms I , D'ALEMBERT. 281 Substitute these values in the above equation, and it will be found to agree with (3) of Art. 405. 408. In considering the writings of D'Alembert on our subject up to the present point, we find but little of importance. Not only do they fail to add anything to what Clairaut had given, but they do not even reach the same level. It seems to me that D'Alembert had not taken the trouble to study a work which far surpassed all his own efforts in the same direction. 409. The next work by D'Alembert is entitled Eecherches sur differens points importans du Systeme du Monde. This work forms three parts or volumes in quarto. The first and the second parts were published in 1754 ; and the third part in 1756. The first part contains the Title, Preliminary Essay, Table of Contents, and Corrections in Ixviii pages; then the text of 260 pages : there is one plate. The second part contains the Title and Table of Contents in vi pages ; then the text of 290 pages : there are three plates. The third part contains the Title, Preface, Table of Contents, and the Privilege du Roi in xlviii pages ; then the text and Cor- rections of 263 pages : there are two plates. 410. There is nothing in the first part with which we are concerned. In the second part we have on pages 201... 209, Eemarques sur la figure de la Terre, gui r&ulte de Id Precession des Equinoxes ; and on pages 265... 290, we have a Chapter entitled De la Figure de la Terre. 411. D'Alembert, on his pages 201 . . . 209, returns to the subject of the information which the theory of the Precession of the Equi- noxes gives with respect to the theory of the Figure of the Earth. He first substantially repeats the matter of which we have given an account in Arts. 385 and 386. He then says, on his page 204 : Je dois cependant avouer qu'un grand Geomctre a cru pouvoir concilier tout, en supposant que la Terre soit im solide Elliptiquc, dont 282 la difference des Axes soit = A > et <l u ^ renferme au-dedans de lui un noyau spherique dont la densite soit a celle du Sphe"roide comme 10 est a 1, et dont le rayon soit au rayon de 1'Equateur comme 3 a 5. D'Alembert here alludes to a memoir by Euler on the Pre- cession of the Equinoxes, published in the Berlin Memoires for 1749 ; see page 315 of the memoir : Euler does not support his suggestion by any theory connected with our subject. D'Alembert shews that the above supposition is inadmissible. Take a formula obtained in Art. 374, namely 4>7rpr 4-7T (a p) s* 4jrpr ' ~3~~ 3r 2 ~T~ let j denote, as usual, the ratio of the centrifugal force at the equator to the attraction there, so that therefore Now let us suppose a- = Wp, and s = - r t so that - -- ^ = 2 very nearly. Thus this value of is smaller than observation will allow. It will be observed that D'Alembert assumes that the ellipticity of the ex- ternal surface is the same as if the outer part were fluid : it is not obvious whether Euler contemplated this in his hypothesis that the Earth consisted of two solid parts. 412. We now pass to pages 265... 290 of the volume. On pages 265... 274, D'Alembert considers how the figure of the Earth D'ALEMBERT. 283 may be found by geographical operations. He suggests in fact that we should assume for the radius vector a series with unknown coefficients involving cosines of multiples of the colatitude. Then by measuring the lengths of degrees of the meridian in various latitudes we find the corresponding values of the radius of curva- ture : and thus we obtain equations for determining the unknown coefficients in the assumed expression for the radius vector. 413. D'Alembert also suggests that observations of the Moon's parallax may be employed for information as to the figure of the Earth: but he admits that practically this method would be of little value. 414. In pages 275... 290, D'Alembert indicates a method for calculating the attraction of a spheroid on a particle at the surface. Suppose Q a point of the surface, G the point which may be called the centre of the spheroid. D'Alembert proposes to con- sider the spheroid as composed of two parts ; one part being the sphere on CQ as radius, and the other part the difference between the sphere and the spheroid. He shews how the approximate value of the attraction of the second part* may be conveniently calculated. It is obvious that the principle of this method is the same as that which has since been developed by Laplace. D'Alembert gives only an outline of his method here ; he works it out in detail in the third volume of the Reclierches...Systme du Monde. We shall recur to it in our Article 424. 415. We now arrive at the third volume of the Eecherches ...Systeme du Monde. Here pages xix...xlii and 107... 260 are devoted to the Figure of the Earth. 416. In pages xix...xlii D'Alembert gives some introductory remarks on the subject, the purport of which is to shew the uncer- tainty as to the actual facts. It was possible to doubt whether the Earth was a figure of revolution ; granting it to be such, it was possible to doubt whether the northern and the southern hemispheres were exactly alike; and granting that they were D'ALEMBEKT. exactly alike, it was possible to doubt whether the figure was that of an ellipsoid of revolution. D'Alembert refers to six measured lengths which had to be considered in testing any theory; live of these were arcs of meridians, namely, those in Lapland, Pern, France, the C 1 ape of (loud Hope, and Italy: one was an arc of longitude, in latitude 4-T o-'. As to a degree of the meridian in France, three lengths had been proposed; Picard gave ~>7(KH) toises ; the Academicians of the North corrected it to .~)71S:> toises; and subsequently it was put at -~>7074 toises: see Art. *2oo'. D'Alembert found it impossible to assign such a value of the ellipticity as would harmonise the six measured lengths. 417. The following points of interest may be noticed in the introductory remarks by D'Alembert. (hi page xxxii lie says that a hemispherical mountain a league high ought to make a pendulum deviate more than 1' from the vertical; but the high mountains in Fern scarcely produced a variation of 7". It is easy to verify his calculation, supposing the density of the mountain equal to the mean density of the Earth. For the faets as to the mountains in Peru see Bouguer's Fiyiirc de la Terre, pages :}<;4...:j<>4. D'Alembert in a note on his page xl suggests, that, in such a mountainous country as Italy, the direction of the plumb-line mav have been disturbed, and thus an error produced in the measured length of a degree. D'Alembert refers to the Pi ure < ':' Jupiter as suggesting by analogy what the figure of the Farth :iiav be; but I do not un- der.-tand all that is said on this matter. The i'ulluwiim' passage O 1 O occurs on pages xxxv and xxxvi. ( 'ar I os observations nous i rouvcnt que la surface dc Jupiter est -iij tic ;i <lcs alterations sans eomparaison ]>lus considerables et ])lus iVequentes quc relic dc ]a 'J'ci'rc ; or si res ;dteruti<ms n'iniliioiont en ricti snr la liLfure <le IY'<|uatcur dc .lupitci', jtounjiioi la figure do lY'(|iia- tciir ile la 'I'efi'e seroit-clle ah'n'e pai- dcs niuuveinens hcauconp lauindre.- .' 1 do not kiiou \\hat change.-, in Jupiter ),< ichr.s to here. D'ALEMBERT. 285 Again he suggests that we should determine by observation whether the figure of Jupiter is precisely that which theory would assign; but I cannot see any practical value in the method which he proposes. He states it thus on his page xli: Pour cela il suffiroit de mesurer le parallele & 1'equateur de Jupiter, qui en seroit eloigne de 60 degres ; si ce parallele se trouvoit sensible- ment egal ou mcgal a la inoitie de 1'equateur, le meridien de Jupiter seroit elliptique ou ne le seroit pas. It seems to me that supposing the observation could be made with great accuracy it would afford but little information ; if the parallel were not exactly half of the equator, we should know that the meridian could not be circular : but we could not in any case pronounce what the figure must be from merely knowing the value of this parallel. 418. We now proceed to the text on pages 107... 260. A brief introduction commences the discussion. D'Alembert proposes to examine the figure of the Earth, first astronomically, so far as observations make it known, and then physically by theory. 419. In the first three Chapters D'Alembert considers whether we can by direct observations determine if certain hypotheses which are usually made are strictly true. Thus, for example, we usually assume that the plane which contains the axis of the Earth and any given place will also contain the vertical line at that place : this amounts practically to assuming that the Earth is a figure of revolution. D'Alembert shews that, strictly speaking, this hypothesis may be untrue; for observations made at any given place would not enable us to decide that the vertical did or did not lie exactly in the plane containing the place and the axis of the Earth. Again, we define the vertical direction at any place as that of falling bodies ; and we know that this direction is perpendicular to the surface of fluid at rest at the place : but this direction will not be necessarily perpendicular to the surface of the solid Earth at the place. Now D'Alembert shews that if the angle between these two directions is very small we shall not bo able to detect it by observations. 286 D'ALEMBERT. I do not give any detailed account of these Chapters, since the propositions are of such a kind that they readily commend them- selves as reasonable. The processes of D'Alembert require atten- tion to understand them ; but they will be found to present no very serious difficulty. 420. D'Alembert's fourth Chapter is entitled De la Figure de la Terre dans les hypotheses ordinaires. This is of the same character as the portion of the second volume of the Becherches which we described in Art, 412. 421. D'Alembert's fifth Chapter is entitled Des parallaxes en tant quelles dependent de la figure de la Terre. The Earth being not a sphere the parallax of the Moon will vary with the place of observation ; D'Alembert investigates formulae for the parallax : but these investigations belong rather to Plane Astro- nomy than to Physical Astronomy. 422. We now pass to D'Alembert's second Section, which is CD titled De la figure de la Terre conside're'e physiquement. 423. The first Chapter, on pages 166. ..177, contains the in- vestigation of certain integrals which will be used in the sequel. Thus, to take the first, required dt (l-f) n n being a positive integer. D'Alemb gral becomes D'Alembert assumes k* + ?k z ? = -- r-; and then the inte- s + l s n ds D'Alembert requires the integral between the limits and 1 1 W of t\ to these limits correspond j^ and for s. He easily obtains the required result by ordinary methods : we will verify by D'ALEMBERT. 287 tfs assuming sin 2 6 = z ^ , which reduces the integral to 1 fc and the value is D'Alembert arrives at the same result on his page 170 ; he ap- parently gives twice this value, but he has really taken the inte- gral twice over. On his page 171, he professes, I think, to investigate the integral dt (l-ff where n is an odd positive integer; but his printing is not very distinct. This integral transforms as before into = f 8*d8 - 2 ^ -*- It is unnecessary for his purpose to take any notice of the nu- merical factor which is here outside the integral sign ; and so he omits it. He gives three times, namely on his pages 174, 176, and 177, the following result : 2 *" x (2rx - a; 2 ) 2 = 16 x 8r 3 (2r*)* ""5x7x9* 424. D'Alembert's second Chapter, on pages 178... 199, is entitled De ^attraction d'un sphe'ro'ide sur les corpuscules places & sa surface ; et de la figure qui en re'sulte pour ce sphe'ro'ide. We begin with a general formula for the attraction of a sphe- roid on a particle at the surface, resolved tangentially ; we shall follow D'Alembert as to principle, but we shall simplify the mere analytical work. 288 D'ALEMBERT. Let there be a point Q on the surface of a spheroid, let s be the distance of Q from a fixed point which we may call the centre of the spheroid ; let 6 be the angular distance of Q from the pole. It is required to find the attraction of the spheroid at Q, resolved tangentially. We assume that the spheroid is a figure of revolution. We may suppose that the spheroid consists of a sphere of radius <?, and an additional shell : see Art. 414. We assume that the shell is at every point so thin that it may be treated as if it were con- densed on the surface of the sphere of radius s. It is obvious that we need only consider the shell when we seek the tangential attraction. Let R be any other point on the surface of the spheroid ; and let its polar co-ordinates be s' and &. Let P be the pole ; put ft for the angle which QR subtends at the centre, and ty for the angle PQR. The element of spherical surface at R may be denoted by s 2 sin fjb dfj, d-*fr ; and thus the element of mass of the shell may be denoted by (s' s) s 2 sin p, dp dty, taking the density as unity. The distance from Q is 2s sin ^ . We first take the resolved part of the attraction along the tangent to QR at Q; and then we resolve this along the tangent to QP at Q. Thus we obtain //. cos | (s 1 s) s z sin ft dp d^r a cos ^r, fa sin D 289 (S - S) COS 2 i COS >|r that is e?//, d>Jr. 2 sin | If we integrate this expression between the limits and IT for jj,, and and 2?r for i/r, we obtain the tangential attraction at Q towards the pole. 425. Now suppose, with D'Alembert, that s' = r +ra (A+B cos & + C cos 2 & + D cos 3 tf'), where a is a very small constant, and r, A y B, Q, D are any con- stants : we might suppose these constants connected by the re- lation A + B+ (7 4- -0 = 0, and then r would be the polar semi- axis of the spheroid. However we will not use this supposition. Substitute this value of s in the expression of the preceding Article : then we see that the tangential attraction reduces to . 27r H 2JOJO sn and cos & = cos 6 cos //, + sin 6 sin /* cos -jr. We shall determine separately the values of the three parts of which the integral is composed. The term involving B reduces to arB rTr2. - I __ sin 6 sin p cos ^ dp d-fy ; Jo Jo this = or B sin 6 TT I cos 3 ^ dp = -5- ar B sin 0. JQ 2 J The term involving C reduces to r r 2ff cos2 | cos ^ ar C I I cos 6 cos JJL sin sin p cos sn /*1T Q this = 2ar (7 sin 6 cos TT I cos 3 ^ cos //, d/z, = - ar C7 sin cos #. J Q 2 5 T. M. A. 19 290 D'ALEMBERT. The term involving D reduces to f"f JOJO (3 cos 2 cos V sin sin //, cos ifr this =ccrZ)sm0cos 2 037r I cos 3 ^cos 2 /^<f/i,+arDsm 3 0^7- cos 3 ^ Jo < 4 Jo ^ ar sin 3 0. Of) dO Thus the whole tangential attraction towards the pole is 47rar [ sin 6 4- sin cos + -^ sin 6 cos 2 + -^ sin 3 ^ ) . \o 5 oo oo / 426. Let there be a solid sphere of radius r and density <r, surrounded by a thin fluid stratum of density <r'; and let the radius of the external surface of this stratum be the s of Art. 425. We propose to enquire if this fluid will remain in a state of rela- tive equilibrium when rotating with uniform angular velocity. The attraction towards the centre may be taken as 5 ; the o resolved part of this tangentially towards the pole is found to the order we require by multiplying by -r-j# , using ^'instead of & in s ci" the result. Let j denote the ratio of the centrifugal force at the equator to the attraction there; then - -^ j sin# cos /row the o pole is the tangential action of the centrifugal force. Thus equat- ing to zero the whole tangential force we get 4?rar ( ^ sin 6 + - sin 6 cos 6 + -^^- sin 6 cos* 6 + -^-- sin 3 6 } &' \o o do oo / o-=0. D'ALEMBERT. 291 Divide by sin 6 ; then equate to zero the coefficients of the various powers of cos 6. Thus we obtain m (3). 3 5? From (3) we get <r = = a ; then from (2) we get a.C= - ; and 7 4 3D then from (1) we get .Z?= -- . D'Alembert has a wrong equation instead of (1), and so his value of B is wrong ; he corrects the error in his Opuscules Maih4- matiques, Vol. VI. page 230. It is remarkable, as D'Alembert says on his page 181, that the value of is independent of B and D, and is numerically the same as it would be if we made cr' = a, and therefore B and D zero, but with the opposite sign. 427. D'Alembert shews that the equation s =r+ ar (A+B cos 6') represents a circle ; supposing a so small that its square may be neglected. He states that on the same supposition the equation s =r + ar (A + B cos & -f C cos 2 0') represents an ellipse. See his pages 181... 183. It is easy to verify these propositions. 428. D'Alembert proceeds to another case of relative equili- brium on his page 183. He first states the value of the attraction towards its centre, produced by an oblatum of small excentricity on an external particle. Suppose the polar semiaxis to be r, and the equatorial semiaxis r (1 + a) where a is very small ; let S be the distance of the attracted particle from the centre of the oblatum, the angle between the polar semiaxis and the direc- tion of 8. Then he says that the value of the attraction towards 192 292 D'ALEMBERT. the centre is 8?rr s a 47rr 6 a 127rr 5 a cos 2 6 he says that this can be obtained by methods given further on, or by other means. We may easily verify this statement. If M be the mass of an oblatum, R the polar semiaxis, e the excentricity ; then the attraction on a particle at the distance from the centre on the polar axis produced, is by Art. 261, approximately M/^ $JR\ Then use the theorem given by Clairaut, Art. 333 ; we have consequently R = r (1 + a sin 2 0), and also E s (1 - e 2 )' 1 = r 3 (1 + a) 2 ; so that e 2 = 2a 3a sin 2 0. With these values of E and e we shall verify D'Alembert's statement. 429. Now suppose the Earth to consist of a solid oblatum of density o-, surrounded by a thin layer of fluid of density or ; as an equivalent supposition we may take two coexistent oblata, the lesser of density a a, and the larger of density <r. Let the polar and equatorial radii of the lesser oblatum be r (1 /9) and r (1 /S) (1 + ') respectively ; and let those of the larger be r and r (1 + a) : we suppose a, a', and ft so small that squares and products may be neglected. Let P denote the gravity of a particle at the pole, and tzr the gravity of a particle at the equator ; the particle being supposed to be on the outer surface. We shall find, by Art. 428, that 167rr(a a') < r -- o - /1X therefore, - =2a+j - (1). D'ALEMBERT. 293 But, by Art. 370, we have fc- ......... (2 ). Substitute in (1) the value of a found from (2) : thus we get Substitute in (3) for a from (2) ; thus = <GT 4 " 2 (10<r - 60-') ' ^ ; ' These results agree with D'Alembert's on his page 186, but the notation is different. It is obvious from (4) that if cr <r and 5j 4a are both positive, then - is greater than ^ ; also if cr - & and oj 4a' are both negative, and 100- 60-' is positive, then - '57 is greater than -j . Also if cr cr' and 5j 4a' are of contrary signs, and 10<r 60-' is positive, then -- is less than -f , 430. It will be observed that the preceding investigation depends on that which we have noticed in Art. 376, and which is not altogether satisfactory, although D'Alembert seems to have been very fond of it. We may also remark that if the layer of fluid is to surround the body completely, there must be a certain condition satisfied, namely, 1 - -ft + a must be less than 1 + a : D'Alembert does not advert to this, but it is not of much im- portance. 431. D'Alembert on his pages 187 and 1 88 makes some remarks on Clairaut. D'Alembert here admits that Clairaut had already obtained the result (3) of Art. 429 ; but D'Alembert says that Clairaut's demonstration was limited to the case in which a is greater than . D'Alembert also states that Clairaut supposed the strata nearer to the centre to be the denser, and also supposed 294 D'ALEMBERT. that oc and a could only differ by a quantity infinitesimal com- pared with a or a!. But these remarks are quite inapplicable. Clairaut believed the strata nearer to the centre to be the denser ; but he did not introduce this belief in such a manner as to restrict his inves- tigations. Clairaut does not limit himself to the case in which a is greater than -~ : D'Alembert seems to have assumed that the quantity denoted by D in Art. 327 is necessarily positive, which it is not. Finally, Clairaut does not assume that the dif- ference between a and a! is infinitesimal compared with a and a', when the fluid is of finite thickness, but only when this thickness is infinitesimal : see Art. 328. D'Alembert certainly added nothing to the investigations given by Clairaut of the theorem which bears his name : in fact, D'Alembert criticised these investigations before he had taken the trouble to understand them. 432. Itis curious to see D'Alembert devote a whole paragraph on his pages 188 and 189 to a very elementary piece of Algebra. If we have given that 12a' (A 1) is greater than 15N (A 1), we must not infer that 12a' is greater than 15JV, unless we know that A 1 is positive. D'Alembert repeats on his page 190 a remark which he had made at an earlier date : see Art. 378. 433. D'Alembert investigates on his pages 191... 197 the values of some definite integrals which are useful in the sequel, [' 2r oc p dx f 2r x f dx namely, various cases of - - = and I - J o n z + Znx+Zrx ' o n*- (n z + Znx+Zrxy ' o (n*-2nx+ 2ne)% obtained by ascribing to p various positive integral values. For example dx 2r J and (n* + 2n.v + 2r*) f (* + ) (2r + n) ' dx 2 n - D'ALEMBERT. 295 We suppose that 2r is greater than n. We observe that the second of these two examples cannot be deduced from the first by changing the sign of n. 434. D'Alembert makes some remarks on his pages 198 and 199 on the attraction of a spherical shell. He takes r for the radius of the shell, and -^- for the attraction on a particle out- side the shell at a distance S from the centre : thus he does not introduce any factor to represent the thickness of the shell. When the particle is inside the shell the attraction is zero. He adds : De-la il me semble qu'on peut conclure que 1'attraction d'une surface spherique sur un point place sur cette surface meme, n'est pas 47r, comme il paroit qu'on 1'a cru jusqu' a present, mais seulement 2-n; If it be necessary to put the idea into words, it would be better to say that the attraction of a .spherical film on a particle which forms part of the film is 2-7T. D'Alembert recurs to the subject of the attraction of a sphe- rical film in the article Gravitation of the original Encydopedie and in the first volume of his Opuscules Math&matiques. 435. D'Alembert illustrates his remarks on the attraction of a spherical film by the following statement on his page 199 : Les Geometres ne sont pas tout-a-fait etrangers a ces sortes de para- doxes, d'une quantite qui s'evanouit tout d'un coup sans disparoitre par degres. Ainsi la courbe y = \fax + *Ja?(b + x) qui est du 8 e degre tant que b n'est pas = 0, perd subitement phisieurs branches lorsque 6 = 0, parce que 1'equation du 8 e degre se reduit alors au 4 e . Yoyez les Memoires de FAcademie de Berlin 1749, page 146. Dans le premier cas, cette courbe a un diametre ; dans le cas de 6 = 0, elle n'en a plus. This illustration does not seem to me very good : it may justly be maintained that the above equation when properly understood is of the 8th degree, even when b = 0. 436. D'Alembert's third Chapter, on pages 200... 213 is entitled Problemes nfaessaires pour gtfntfraliser les recherches precedentes. This Chapter consists of various definite integrals which are required by D'Alembert in his process for calculating the attraction of a spheroid. These definite integrals depend 29G D'ALEMBERT. rnainlv on the alues of I - - , when for ;> we put in J r (tf + r'-^M) 1 succession 0, 1, 2. o, 4, .">, (>. Here S ami r arc constants: the integrals present different values according a.s B is greater or less than r. D'Alembert puzzles liis readers by taking 1 and 1 as the limits of u on his pages '201, 202, and 20S ; but except on these pages the limits are those which I have stated, namelv r and r. His results are correct, allowing for a few obvious misprints. 437. D'Alembert's fourth Chapter, on pages 214...24G is entitled Usages dcs Problemes prfae'dens, pour determiner V attraction du sphtro'ide sur un corpuscule quelconque. He determines the attraction which a certain spheroid of revolution exerts on a particle, external or internal, at right angles to the radius vector, and along the radius vector; these he calls respectively the horizontal and vertical attractions. He states the results, having previously given the values of certain definite integrals which a*e required. We will explain how these results may be verified ; the method we shall adopt is that which we have already used in Art. 424. IVt C denote the centre of the spheroid, CP the semi-axis of revolution, (J any point on the surface having for its polar co- ordinates ,s and 0. Produce C<J to any point <j. It is required to find the horizontal attraction on a particle at y. Let C<j[ 3. D'ALEMBERT. 297 Let H be any point on the surface having for its polar co- ordinates s' and 6 '. We suppose that the spheroid consists of a sphere of radius s and an additional shell. Let the angle PQR be denoted by ^, and the angle QCR by /A. The element of the shell at R s 2 (s' s) sin p dp dty. The distance Rq = (s 2 + S 2 - 2s S cos /LI)*. The resolved attraction of the element in the plane RCq at right angles to Cq is therefore s' sin p s' 2 (s f s) sin p dp d^r t ~~ and resolving along the. plane PCq we get s sin p cos ^r s* (s' s) sin p dp d^r We have to integrate this between the limits and TT for p, and and 2?r for -\Jr; then we obtain the horizontal attraction at q towards P. We suppose with D'Alembert that s' has the value given in Art. 425. We shall obtain by effecting the integrations, neglecting the square of a, 8 Or* 47V 12Z)r 6 IWr* In like manner if q be between C and Q instead of on CQ produced, and Cq be called 8 as before, we obtain for the horizontal attraction 8CS Trasin ^ o u u oor t r ) These expressions must be multiplied by a factor to represent the density, if the density is not unity. When 8 = r these expressions both coincide, as they should do, with that given in Art. 425. 298 D'ALEMBERT. 438. The attraction at q in the direction at right angles to the meridian plane of q will be zero, since the spheroid is supposed a figure of revolution. D'Alembert himself makes the remark on his page 216. He adds however that this can also be seen by calculation ; and he gives some calculations, which I do not find to be intelligible. 439. In Art. 437 we have investigated expressions for the horizontal attraction of the spheroid supposed homogeneous. D'Alembert deduces on his page 218 the attraction of such a spheroid on an included particle when the spheroid is composed of indefinitely thin shells of varying density : the process is the same as we have already found was used by Clairaut. See Arts. 323 and 336. 440. In order to obtain the whole action along the tangent to the meridian curve at any point, we must as in Art. 426 add to the horizontal attraction the resolved part of the vertical attraction along the tangent, and also the resolved part of the centrifugal force. 441. Next we proceed to find the vertical attraction on the particle at q. Suppose the particle outside the spheroid. The vertical action of the sphere of radius s (1 + 3a A + 3xB cos 6 + 3.0 cos 2 6 + 3aZ> cos 8 0}. We must now determine the vertical action of the shell. .As in Art. 437 we find that this is / Jo J 27r (3 s' cos fj,) s* (s s) sin p dp d^r By effecting the integrations we obtain for the whole vertical attraction f/) + TTOL cos 6 D'ALEMBERT. 299 In like manner if the attracted particle be inside the spheroid, the whole vertical attraction is , Tra cos 6 lo f 4#> Wr j_.__ .__ Try. cos 3 /, ?ra cos = Ir For a point on the surface we must put in either of these results 8 = r (1 + a A + aB cos + aC cos 2 6 + otZ> cos 3 0) : it will be found that each of them becomes then 4-rrr /, 2x<7\ /i&Dr /i4<7r ./,8Dr 1 + oc^4 H = + Tra cos - * Tra cos 6 - Tra cos 6 -^- . 3 \ o / 3o lo 21 These expressions must be multiplied by a factor to represent the density, if the density is not unity. Then as in Art. 439 we can obtain the vertical attraction for a spheroid composed of indefinitely thin strata of varying density. 442. D'Alembert now discusses the relative equilibrium of homogeneous fluid surrounding a solid nucleus composed of strata of varying density: see his page 222. The problem is thus an extension of that in Art. 426, and it is solved in the same manner. There is no difficulty in the hydrostatical part of the problem ; for since the fluid is homogeneous it is sufficient for equilibrium that the tangential action at every point of the surface should be zero. If we equate to zero this tangential action, we obtain a result of the form sin 6 (M Q + M^ cos 9 + M a cos 2 6) = 0, where J/ , M lt and J/ 2 are independent of 6. This leads, as in Art. 426, to three equations J/ = 0, If 1 = 0, Jf a = 0. D'Alembert gives these three equations on his pages 222... 225. 443. We must be careful as to the notation, since many sym- bols are required. D'Alembert leaves his notation to explain itself, and it is not very inviting. I shall use the subscript 1 to denote values relating to the external boundaiy of the fluid ; and I shall use p as a general symbol for the density. Then the three 300 D'ALEMBERT. equations are , p J , i ' , A - These equations may be developed. I shall use the subscript to denote values relating to the internal boundary of the fluid. Then between the limits r and r x the density is constant, by hypothesis ; I shall denote it by cr. Also for 8 we must put r x to the order which we wish to retain. I will now express the second of the three equations in the modified form which arises from the use of the notation just explained ; the other two equations may be similarly expressed. In this equation G> denotes the angular velocity ; and I shall (O^T put as usual j for * . Thus we have J 4?r r ?r / r i 2 , t pr*d i Jo This equation corresponds with D'Alembert's on page 225; he puts it so as to express a in terms of the other quantities. He takes 7^ = 1 and 0^=1, which he may do; but then by mistake or misprint he also takes C = 1, which he ought not to do. This equation also exactly corresponds with equation (3) in Art. 324 ; the a.C of the present Article is the e of that Article. 444. D'Alembert passes on his page 225 to the problem in which the entire spheroid is fluid, and is composed of indefinitely thin strata of varying densities. He treats this problem according D'ALEMBERT. 301 to his own peculiar views of hydrostatical principles. He arrives at three general equations, each of which presents itself in a primary and in a derived form, like Clairaut's equation of Arts. 341 and 343. D'Alembert's peculiar views lead him astray, and the con- sequence is what we have already seen in Art. 405, namely, that the results which he obtaios are much more complicated than they should have been. For instance the third equation, with the notation of Art. 405, is presented thus in its derived form by D'Alembert : 2 ^>r 1 D r * N d (I d^fp dr\ ] r* T(f)P " T(r) dr (r* cfc(f dp) j ' where N is a constant. But the correct form is that in which the right-hand member is zero. His second equation is precisely the same as (5) of Art. 405, with C instead of e ; the error and the correction are the same as we have already indicated in that Article. In like manner the derived form of D'Alembert's first equation is similarly embarrassed with a superfluous term. The K which occurs on his page 231 should be zero. D'Alembert admitted his errors in the fifth volume of his Opuscules MatMmatiques, page 5. 445. These differential equations for C and D, when written correctly with zero on the right-hand side, are case's of the general equation, which Laplace's functions must satisfy, in Laplace's Theory of the Figure of the Earth. This general equation is , + v dr 2 r* V- 2 dr " ^'p 3 ~7^ j~ If we put 2 for i we arrive at the same differential equation for F 2 as for D'Alembert's symbol (7; and if we put 3 for i we arrive at the same differential equation for F 3 as for D'Alembert's symbol D. Laplace shews that F 3 must be zero. If we put D = in the differential equation for D'Alembert's symbol B, we find that this is the same as the above when 1 is put for i. 302 D'ALEMBERT. 446. D'Alembert makes some remarks on the integration of the differential equations which have been obtained ; see his pages 231... 234. By transformation he arrives at an equation which he says is integrable in several cases ; he gives three cases : they are however unintelligible to me. 447. On his pages 234... 244 D'Alembert extends the calcu- lation of the horizontal and vertical attractions which we have noticed in Arts. 437... 441 : he introduces two new terms into the expression for the radius vector of the attracting body, namely, I have found on going over the calculation that there are numerous misprints or errors in his results. 448. On his page 245 D'Alembert takes the case of a sphe- roid composed of two fluids of different densities ; he says that the figures of the upper and lower strata must be determined by the law of the perpendicularity of the action to each of the strata. He adds : Car dans le cas ou les couches voisiues different entr'ellea sensible- men t par la densite", et ont une epaisseur finie, la pesanteur doit etre perpendiculaire a chacune. Voyez 1'Appendice de mon Essai sur la resis- tance desfluides. The statement he makes here about the conditions of equili- brium is true ; but the reference to the Essai sur la resistance des Fluides is very, remarkable : for the doctrine maintained in the Essai is precisely the reverse of that which is here affirmed in the Recherches. We read in the Essai on page 206 : Supposons maintenant que le Fluide soit compose de plusieurs couches differemment denses, et dont la difference de densites soit finie ; je dis que le Fluide pourra encore etre en equilibre, quoique les surfaces qui separent ces differentes couches ne soient point de niveau... Suppose p the pressure at any point of the surface bounding fluids of different densities. Let S be the force, if any, resolved along a tangent to the surface. Then proceeding along an element of this tangent we should have in one fluid dp = pSds, and in the other fluid dp = p'Sds, where p and p are unequal. But these values of dp must be equal ; therefore S must = 0. D'ALEMBERT. 303 This assumes that there is no discontinuity in the forces acting at the common surface. In the remarks on page 206, D'Alembert's Essai, which follow and support the words we have quoted, he allows a discontinuity to occur in the forces. 449. D'Alembert's fifth Chapter, on pages 247... 260 is enti- tled De I attraction dun sphe'ro'ide qui nest pas un solide de revolution. This Chapter is not important; it merely indicates how we ought to proceed, and shews that in some cases the inte- grations could be effected. 450. On his page 256 he makes a mistake to which I have drawn attention in Art. 381. He says : J'ai fait voir, par exemple, dans mes Recherches sur la cause des vents art. 84. n. 10. qu'uu sphero'ide elliptique, homogene et fluide, tournant autour de son axe, ne pouvoit subsister, si les meridiens n'etoient pas tous egaux et semblables ; . . . 451. On his page 258 he alludes to the case in which we require the attraction, not of a whole spheroid, but of a segment of a spheroid. Then on his page 259 he takes for special con- sideration the case of a semi-spheroid ; but his first paragraph is unintelligible to me : in his second paragraph he asserts that the attraction along the radius of a semi-spheroid is half the attraction of the whole spheroid, which, however, is not necessarily true of any semi-spheroid, though it would be true if the whole spheroid were cut symmetrically into two halves. 452. Let us now appreciate the contributions to our subject which D'Alembert made in his Recherclies...Systme du Monde. The method of estimating the attraction of a spheroid by re- solving the body into a sphere and a thin additional shell, which is here systematically employed, is very valuable. Assuming that the radius vector of a spheroid is r + ar (A + B cos & + C cos 2 & + D cos 3 ff + E cos 4 ff + F cos 6 ff) where a is very small, he gives expressions for the resolved at- tractions on any particle, external or internal, the spheroid being either homogeneous or composed of indefinitely thin strata of 304 D'ALEMBERT. varying density. The calculations are laborious; and though D'Alembert's results are not free from error, yet they furnish useful information. Retaining the terms in the radius vector as far as D cos 3 & inclusive, D'Alembert gave the equations which must be satisfied by B, C, D, supposed variable, to ensure the relative equilibrium of a fluid mass. His equations are encumbered with terms which are really non-existent ; but still in their derived forms the re- markable similarity between them to which we have drawn atten- tion in Art. 445 is 'made apparent. I consider it to be quite possible that this similarity may have struck the attention of Legendre and Laplace, and thus contributed to the construction of the general equation. As I have already hinted, D'Alembert himself over estimated the value of the conclusions that he drew from his peculiar views of Hydrostatics. In the preface to the third volume of these Recherches . . . . Systeme du Monde, page xxxvi, he states that hitherto the Theory of the Figure of the Earth had been re- stricted to verifying the agreement of the elliptic figure with the laws of Hydrostatics ; and then adds, " j'ai trouve de plus, et je le de'montre dans cet Ouvrage, qu'il y a une infinite' d'autres figures qui s'accordent avec ces loix, surtout si on ne suppose pas la Terre entierement homogene." This, however, as we now know is unsatisfactory. For instance, D'Alembert indeed arrives at an equation which his symbol D must satisfy, as we saw in Art. 444 ; but he does not solve the equation, and so shew that D is a real quantity: on the contrary, Laplace, in fact, shews that D must be zero. CHAPTER XIV, BOSOOVICH AND STAY. 453. THE present Chapter will contain an account of the contributions made by Boscovich to our subject, together with a notice of the poem by Stay to which Boscovich added copious explanations. 454. In 1750 two Jesuits, Maire and Boscovich, "began to measure an arc of the meridian in the Papal States. The account of the survey appeared at Rome in 1755, under the title De Litteraria Expeditions per Pontificiam Ditionem. The volume is in quarto ; it consists of Title, Dedication, Preface, and Index in xxii pages, and the text in 516 pages: there are three pages of Errata, and four Plates. A French translation was published at Paris in 1770. The dedication is to Benedict XIV., by whose command the survey was executed : behind the cloud of incense raised by the authors, we may discern the figure of a sagacious and enlightened Pontiff. 455. The book is divided into five parts. The first gives the history of the proceedings, the second the calculations for the determination of the length of a degree of the meridian, the third the correction of the map of the district, the fourth an account of the instruments employed, the fifth a treatise on the Figure of the Earth. The second and third parts are by Maire ; the others are by Boscovich. I shall not enter into any examination of the practical opera- tions recorded in the volume; they have been criticised by T. M. A. 20 306 BOSCOVICH. De Zach in his Correspondence Astronomique, Vol. VI. : see how- ever, Airy's Article on the Figure of the Earth, in the Encyclo- paedia Metropolitan^ page 207. 456. The fifth part of the book is that which we have to examine, This occupies pages 385... 516, and is entitled De Figura Telluris determinanda ex cequilifoio, et ex mensura gra- duum. After a few introductory sentences, the treatise is divided into two Chapters : the first, extending to page 481, relates to the Figure of the Earth, as deduced from the theory of fluid equi- librium ; the second relates to the Figure of the Earth as deter- mined by the measure of degrees. 457. It must be observed that before the publication of the book, Boscovich had issued various dissertations, bearing more or less on our subject : these seem to have been academical exer- cises which he delivered in his character of professor at the Roman College. I have not seen any of these dissertations. Boscovich refers to them generally on pages xviii. and 386 of the book : from the latter page it appears that few copies of the exercises were printed, and of these the larger part perished. Probably the treatise reproduces all that was valuable with re- spect to our subject in the previous publications. The dates and titles of some of these dissertations are given in the pages of the work which I have recorded after them : 1738. De Telluris figura, 23. 1739. De figura Telluris, 395, 399, 445, 447, 487. Perhaps we may infer from the last three lines on page 445, that this was reprinted in a subsequent year. 1741. De Inasqualitate Gravitatis, 23. 1742. De Observationibus Astronomicis, 23, 475. 1748. De Maris ^Estu, 390. De Lege virium in natura existentium, 416. The date is not stated, but it is said exposui nuper. I give the titles as I find them : it is possible however, that there may be only one dissertation instead of the two which ap- pear dated 1738 and 1739. BOSCOVICH. 307 458. The work of Boscovich, to which we now proceed, may be described in general as out of date, even when first published : it is chiefly written in an antiquated geometrical fashion, which one would have thought little likely to be adopted for this subject after the appearance of Clairaut's treatise. The Latin seems to me much more elaborate than is usual in the scientific litera- ture of the period : this might perhaps have been expected from an Italian and a Jesuit. 459. Up to his page 417, Boscovich considers the case of homo- geneous fluid attracted to a fixed point by a force which is any function of the distance, and rotating with uniform angular ve- locity round an axis through the fixed point : the analytical solu- tion of this problem is very short and simple, as we have seen in Art. 56. Boscovich gives correct but tedious geometrical construe - tions, and devotes special attention to two cases, namely, that in which the force is constant, and that in which it varies as the distance : in this way he contrives to fill thirty pages. Boscovich gives on his page 399 a good elementary investiga- tion like that on Clairaut's page 143 : see Art. 297. 460. A strange mistake occurs on pages 411 and 412, Bos- covich has assumed a value for the radius of the equator, and has found as usual that the ratio of the centrifugal force at the equator to the attraction there, is that of 1 to 289. He adds : Si gradus sequatoris fuerit major, vel minor, in eadem ratione duplicata major, vel minor erit sinus versus amis similis, adeoque et vis centrifuga, et proinde in eadem ratione duplicata minuendus erit posterior proportionis numerus. But the word duplicata ought to be omitted : moreover, cor- responding to the words vel minor, the words vel augendus should be inserted after minuendus. 461. On his page 417, Boscovich says that he will investigate the Figure of the Earth on the Newtonian hypothesis of gravity, and will illustrate in the first place Maclaurin's solution: Boscovich refers to Maclaurin's Prize Essay on the Tides, and not to the more complete investigations contained in the Fluxions. 202 308 BOSCOVICH. Thus from page 417 to page 448, Boscovich may be said to reproduce in substance Maclaurin's discussion of the relative equilibrium of a mass of homogeneous rotating fluid. We shall only have to notice a few matters which present some novelty. 462. Boscovich begins with a demonstration of a theorem in Conic Sections which forms the fourth Corollary to the first Lemma in Maclaurin's Essay. Boscovich considers his own demonstration, which is geometrical, more simple and more ele- gant than an analytical demonstration, which he ascribes to Calandrinus, printed in what we call the Jesuits' edition of the Principia. Boscovich does not remark that Clairaut had already given a very good demonstration by the method of projections: see Clairaut's page 159, 463. On his page 424, Boscovich enunciates the following theorem : Si in massa quadam fluida particular omnes ejusmodi viribus animate sint, ut assumpto intra earn puncto quocumque, bini quicumque canales rectilinei ducti inde ad supei ficiem extimam in aequtlibrio sint, ea inassa erit in sequilibrio. In his demonstration he shews that if at every point rectilinear columns are in equilibrium, so also are curvilinear canals of every form, and that a particle at the surface has no tendency to move. The part relating to curvilinear canals is the most interesting: this, however, had already been formally treated by D'Alembert in his Essai sur la Resistance des Fluides, page 15. On his page 432, Boscovich supplies in fact a demonstration of what Maclaurin contented himself with affirming in the words " in like manner it is shewn": see Art. 245. 464. Boscovich has to compare the attractions of an oblatum on a particle at the pole and at the equator respectively. After remarking on page 435, that Newton had shewn how to calculate the attraction at the pole, Boscovich adds : sed pro puncto posito in aBquatore rem nequaquam perfecit, vernm crassa quadam sestimatione invenit utcumque pro ellipsoide data, BOSCOVICH. et parum abludente a sphaera. Mac-Lavrinus multo sane elegantius accuratissiiue, et felicissime rein perfecit tarn pro puucto posito in polo, quam pro puucto posito in sequatore ;... However, Boscovich says that he will himself adopt a method which is nearly the same as Bernoulli's ; it is the method, really due to Clairaut, which we have noticed in Arts. 1G5 and 233. Boscovich professes to use Geometry alone: but the Geometry consists chiefly in denoting the length of every straight line by two capital letters instead of a single small letter: this strange notion of Geometry has survived to our own times in the Uni- versity of Cambridge. 465. Boscovich arrives at the usual result for the attraction of the excess of an oblatum over the inscribed sphere, on a particle at the pole ; and with some enthusiasm he says on his page 438, Et eaquidem est elegantissima, et sirnplicissima expressio ejus vis. From this result he deduces very briefly and easily the attrac- tion of the excess of a sphere over the inscribed oblatum on a particle at the equator : see his page 439. Hence finally he arrives at the equation which we have fre- quently given in our notation ; namely e = - : see his page 441. 466. A digression on pages 442, ..447 is devoted by Boscovich to Hermann. I have already noticed Hermann's Phoronomia, and I presume this is the work Boscovich has in view ; but it does not seem so obvious to me as to Boscovich, that Hermann held New- ton and David Gregory to be wrong : see Art. 95. Boscovich says on his page 442 : Et quidem Hermannus censuit, hanc ipsam snam EllipFhn esse illam,. qure in Newtoniana gravitatis theoria debeat obvenire, ac Gregorium, et Nevvtonum ipsum culpandos existiinavit, quod ii id ipsum non viderint, et plusquam duplo majorem justo compression ern Telluri tribuerint, quam ipsa illorum principia postularent. At Hermannus ipse in eo erravit sane quamplurimum,... The ipsum after Newtonum marks Boscovich's opinion of Her- mann's audacity. 310 BOSCOVICH. The digression is interesting because Boscovich allows that he was himself for a time to some extent misled by Hermann. Bos- covich in 1739 was thus induced to suspect that the oblatum, which Newton had assumed without demonstration, was not a possible form for relative equilibrium; but in the following year Maclaurin's demonstration settled the matter, and then Boscovich was led to investigate the cause of Hermann's error : accordingly he points out what he considers to be the erroris primus fons, and the alter ejusdem erroris fons. It may be doubted whether Boscovich himself was quite clear on the subject ; he appears to fall into the mistake which has been pointed out in Art. 33, for he does not introduce the important condition involved in the words resolved along tlie radius: see his page 443. But in his commentary on Stay's poem, at a rather later date, he is quite correct : see his Article 244 on pages 371 and 372 of Vol. n. 467. On his page 448, Boscovich gives an elegant investiga- tion of the diminution of gravity in passing from the pole to the equator. But by gravity he really means gravity resolved along the radius, which is not strictly the same as the gravity which is measured by observations : see Art. 34. 468. Boscovich now proceeds to consider the Figure of the Earth when it is not supposed to be homogeneous. He assumes that there is a spherical homogeneous nucleus surrounded by fluid which is also homogeneous, but not of the same density as the nucleus : to this discussion he devotes his pages 448... 457. The investigation is tedious, but was probably considered by the author to be a choice specimen of his geometrical methods. Although the whole discussion was quite superfluous after the publication of Clairaut's treatise, yet there is one matter of principle in which Boscovich is rather superior to Maclaurin. As we have already stated, when Maclaurin supposed the earth to be fluid, but not homogeneous, he did not demonstrate that the whole mass would be in equilibrium ; see Arts. 264, 267, 269. Boscovich shews by his language that he saw this difficulty ; he says on his page 458 : BOSCOVICH. 311 Idcirco ego, nt metliodum canalium tuto adhiberera, massam solidam prius ad homogeneitatem adduxi, amandata in centrum redun- dante inateria, turn dissolvi. We will give a notion of Boscovich's method. Suppose we have to consider the case in which there is a homogeneous fluid surrounding a solid spherical nucleus ; and let the density in the nucleus be a function of the distance from the centre. Reduce the density of the nucleus to that of the fluid, and put a force at the centre, producing an attraction equal to that of the excess of the solid nucleus above an equal volume of fluid. Then suppose the nucleus to become fluid. If the additional force at the centre attracted as the distance from the centre, we should thus obtain a problem which has been fully discussed by Maclaurin ; for he has considered forces varying as the distance from the axis and from the plane of the equator, besides the attraction of the fluid : see Art. 245. Boscovich then by a supplementary investigation has to allow for the difference between his supposed force at the centre which attracts as the distance, and the real force which would attract inversely as the square of the distance. 469. Boscovich obtains a result which, as he says, had previously been given by D'Alembert and Clairaut : see Art. 377. Boscovich points out that the result differs from one which Daniel Bernoulli had given in his Essay on the Tides, and which had been criticised by D'Alembert. I do not stay to discuss the point, as it does not strictly belong to our subject but to that of the Tides. See D'Alembert's Reflexions sur...des Vents, page 56; Laplace's Mtcanique Celeste, Vol. v. page 150; and page 8 of Lubbock's work mentioned in Art. 233. It may however be observed that Boscovich seems to have supposed that D. Bernoulli's result ought to have coincided with his own, although the circumstances of the problems differ in a very important respect. In D. Bernoulli's problem the fluid is exposed to the attraction of a distant body, and this attraction does not reducfe to a single force tending to the centre and varying as the distance, which is the case that Boscovich con- siders. BOSCOVICH. 470. In his pages 459... 465 Boscovich discusses the result which, as we have stated in Art. 469, he had obtained in agree- ment with D'Alembert and Clairaut. Boscovich shews that in certain cases the external surface is an oblongum, not an oblatum; it appears however from his page 463, that he held the oblongum to be in modern language an unstable form. See Art. 378. 471. In his pages 466. . .468, Boscovich demonstrates Clairaufs Theorem, on the same hypotheses as to the constitution of the Earth which had been used to obtain the result of Art. 469. He draws some inferences from the theorem in his pages 469... 471. 472. Boscovich now proceeds to the subject of the variation: of gravity as tested by experiments with the pendulum. He sug- gests local inequalities as the cause of the observed irregularities. He calculates the effect which would be produced on the plumb- line by the attraction of a sphere of the mean density of the Earth, of a geographical mile in radius, for various positions of the sphere ; see his pages 472... 474. One of his results is that such a sphere as we have mentioned, if placed just beneath the surface of the Earth in addition to the matter already there, would increase the length of the pendulum by one-eighth of a line. Then he says that if for the depth of eight geographical miles the density at the pole is twice that at the equator the length of the pendulum at the pole will be a line longer. This, he says, follows from what has been demonstrated : but there seems to be some mistake. If r be the radius, and p the density,, of a sphere, the attraction at the surface is . o Now if the density at the pole is changed from p to 2/o throughout the depth h, the additional result is approximately the same as would be produced by the attraction of an infinite plate of thick- ness h : and so it is Zirph. Suppose h = 8r ; then the result be- comes IQirph: this is twelve times the former result ~ . Ac- o eordingly instead of an increase of one line in the length of the 12 8 12 pendulum we obtain an increase of -^ of a line, that is, of a line and a half, * BOSCOVICH. 313 473. Boscovich refers to the curious opinion expressed by Newton to which we drew attention in Art. 31 ; Boscovich says on his page 475 : Newtonus censuit prope sequatorem debere densitatem esse potius majorem in partibus nimirum. a Sole quodaramodo veluti tostis. Ego contra, cum tarn multa corpora dilatentur caloris vi, et vi frigoris adstringantur, opinor debere potius rariora ibi esse corpora ob id ipsum. Sed externi caloris, et frigoris vis ad tantam altitudinem infra supe'rficiem non pertingit, ut effectual sensibilem edat in partem utramlibet. 474. Boscovich notices the fact that according to observations made by Bouguer and La Condamine, the attraction of a large mountain in Peru was much less than it ought to have been, supposing its density equal to the mean density of the Earth : see Art. 363. Boscovich offers a conjecture in explanation of this fact ; he says on his page 475 : Verum montes quidem plerique, ut ego arbitror, effecti sunt intumescentibus interni caloris vi stratis superficiei proximis ; quod quidem si ita contigit, nihil ibi materise accedit, et vacuus inter viscera hiatus compensat omnem illam apparentem niaierise in montem assur- gentis cougeriem. 475. Boscovieh observes that a greater effect might be pro- duced on the pendulum by a large tract of raised land than by a single mountain. He refers to a problem on this point which he had given in his dissertation De Observationibus Astronomicis, 1742. The problem is the following : cut a slice from a sphere by two parallel planes, one passing through the centre ; bisect the slice by a plane perpendicular to the circular ends: then find the attraction, resolved parallel to the planes of the circular ends, of one of the halves on a particle situated at that point of the half which was originally the centre of the sphere. Boscovich states, without investigation, an approximate result for the case in which the thickness of the slice is very small compared with the radius of the sphere : but this result is incorrect. In his commentary on Stay's poem, Vol. II., page 382, he gives a correct investigation. If we wish to confine ourselves to the order of approximation which is sufficient for his numerical application, we may replace the slice of a sphere by the slice of a cylinder. Let a 314 BOSCOVICH. * be the radius of the slice of the cylinder, h the thickness, p the density. Then the required result is easily found to be dr o , , a o that is 2ph log - ~ If we suppose h very small compared with a, we get approximately 2ph log -7- , that is ph 1 2 log 7 -f 2 log 2 J ; this agrees closely with pk ( 2 log 7 + 1*389 J , which is Boscovich's result in his Commen- tary on Stay's poem. 476. Boscovich makes a curious suggestion on his page 477. He proposes to have a pendulum in a tower by the sea shore, at some place in England or the opposite continent, where the water may be raised by the high tide 50 feet above the level of the low tide. He considers that if the density of the sea is equal to the mean density of the Earth, a deviation of about 2" will be produced in the direction of the pendulum. By having a long pendulum and using a microscope, he thinks the deviation might be observed, and thus some notion obtained of the mean density of the Earth. See some remarks on this suggestion in De Zach's work, L' attraction des montagnes, page 17. 477. In his pages 477... 481, Boscovich cites some observa- tions of pendulums, and draws inferences from them : he had recently made some observations at Rome, in conjunction with La Condamine, with the pendulum which had been used in America and at the Cape of Good Hope. 478. We now reach the second Chapter of Boscovich's treatise ; this relates to the Figure of the Earth, as determined by the measure of degrees. He begins with some general explanation as to what is meant by a degree, and an osculating circle; see his pages 481... 486 : these present nothing of interest except a curious mistake. Let s denote an arc of a curve measured from some fixed point, p the radius of curvature at the variable extremity of the arc, and <j> the inclina- BOSCOVICH. 315 tion of p to a fixed straight line : then we know that ^ = p. By the length of a degree, we mean the value of lpd<j) taken between limits which differ by the circular measure of a degree. Thus the length must be equal to p l ^-^ , where p l is some value of p which lies between the least and the greatest of the values which occur within the range of integration, p being supposed always finite. This statement follows from the first principles of the Integral Calculus ; Boscovich, however, denies the universal truth of it, for he says on his page 484 : ..... Fieri itidein potest, ut arcus unius gradus plurimum differat a gradu circuli osculantis curvam ubique intra eum arcum, quod quidem turn accidere potest, cum curvatura pergendo ab altero ejus extremo ad alterum primo quidem perpetuo crescit, turn perpetuo decrescit, vel vice versa. 479. On his page 487, Boscovich seems to adopt a definition which has not been used by others. If 2a and 26 are the major 26 2 and minor axes respectively of an ellipse, we call - - the latus. 26 2 rectum : Boscovich seems to call the latus rectum with respect a 2a* to the major axis, and -j- the latus rectum with respect to the minor axis. In his pages 487... 493, Boscovich gives various geometrical constructions relating to the ellipse and its radius of curvature ; he says on page 488 : Exhibebo autem solutiones diversas ab iis, quas simplicissimas sane, et admodum elegantes, ac geometricas itidem exhibui in mea dissertatione ilia de Figura Telluris. Thus he seems to have been very well pleased with some of his own work ; for I presume we are to consider the demonstra- tions in the book at least as good as those which had appeared in the dissertation. 316 BOSCOVICH. A property of the ellipse may be noticed which he demonstrates on his page 489. The normal at any point P of an ellipse meets the minor axis at G ; from P a perpendicular PM is drawn to the minor axis meeting at Q the circle which is described on the minor axis as diameter; from G a straight line is drawn parallel to CQ, meeting MP produced at N \ then GN is equal to half the latus rectum with respect to the minor axis, and MP is <i mean proportional between MQ and MN. 480. Boscovich obtains on his page 494 an approximate formula, which determines the ellipticity of the Earth from the lengths of a degree of the meridian at the pole and the equator ; Boscovich refers to Maupertuis, who had previously obtained the formula: see Maupertuis's Figure de la Terre...page 130. Boscovich however considers that the exact theorem is. more elegant, namely, that the lengths of a degree of the meridian at the equator and the pole are respectively as the inverse cubes of the corresponding diameters. Boscovich shews on his pages 495 and 496 that the diminu- tion of the length of a degree of the meridian from the pole to the equator varies as the square of the cosine of the latitude : hence the ellipticity may be found by measuring arcs of the meridian. 481. Boscovich now proceeds to consider the actual measures of a degree of the meridian in various places. He says that there are only five measures which are accurate; namely, those in France, in Lapland, in Peru, at the Cape of Good Hope, and his own in the Papal States : see his page 4.97. He holds that the value of Picard's degree may now be con- sidered perfectly settled post mutationes quatuor. It is not certain what is meant by four changes; in Art. 236, four different values are given, and these are also recorded by Boscovich himself in his commentary on Stay's poem, Vol. II., page 392. But if there were four changes, there must have been five different values : perhaps then we are to include a result obtained by J. Cassini, which was between 30 and 50 toises less than Picard's own : see De la Grandeur et de la Figure de la Terre, page 286. BOSCOVICH. 317 Boscovich alludes to Norwood's measure, and gives a few lines to Snell's measure ; he considers them both unsatisfactory : see Arts. G8 and 105. 482. Boscovich on his pages 499... 503 deduces the ellip- ticity of the Earth by ten different binary combinations of the five arcs ; but he finds that the results are very discordant. One combination actually brings out a negative ellipticity ; namely, the combination of the Roman arc with the African. The other com- binations give various values of the ellipticity, the greatest being , and the least about one-tenth of this. The mean J.Zo ellipticity is ; ; but if the two combinations be rejected which ADD differ very much from the rest, the mean ellipticity is . J yo Boscovich has some troublesome misprints on his page 501; the ellipticities deduced from his sixth and tenth combinations are quite wrong: and the numbers which he gives in his following Article to denote the mean excesses of the polar degree above the equatorial are a third of the true values, 483. Boscovich says on his page 501 that some persons had tried to conciliate the results by forcing the observations : Nonnulli, ut nuperrime Eulerus in schediasmate, cujus summam quanclam mihi humanissime cornmunicavit hie Romae praesens, dum hsec scribo, Condaminius, observation ibus vim inferunt, ut onrnia concilient. Et is quidem gradum Lapponiensem, Africanum, Quitensem, mutatione adhibita hexapedarum 19 in singulis, conciliat cum ellipsi Newtoniana, sed Gallicus Piccardi gradus corrigendus illi est hexapedis 169, quern idcirco sibi maxiine suspectum esse profitetur, et novas in Gallia niensu- ras desiderat. At id quidem errorem exposcit intolerabilem sane in gradu cum ingenti cura definito a peritissimis viris. It seems absurd to suppose that an error extravagantly the greatest should occur in the arc which must have been the best determined of all at the epoch. We shall recur to Euler's specula- tions in Chapter XV. 318 BOSCOVICH. 484. On his pages 502... 506, Boscovich discusses Bouguer's hypothesis that the increment of the length of a degree of the meridian in proceeding from the equator to the pole varies as the fourth power of the sine of the latitude : see Art. 363. Boscovich considers that the African arc overturned this hypothesis. But then it should be observed that the African arc presented much difficulty when compared with the others. Boscovich observes in a despairing tone : " Quocumque te vertas, nihil certum, sibi constans, et regulare occurrit. " He gives on his pages 507... 510 reflections on the state of knowledge of the subject : he sums up his opinions vigorously on his page 508 as to what had been established. Instead of the inquiry respecting the Figure of the Earth from the measures of degrees being finished, he considers that it had scarcely been commenced. Still some valuable results had been obtained : the hypothesis of an attraction directed to a fixed point was excluded, and the compression at the poles was extremely probable, though the amount of this compression was uncertain. 485. On his page 510, Boscovich says that the observations were not inconsistent with the hypothesis of a nucleus in which the density, in modern language, is a function of the distance from the centre. He makes two statements, as to what Clairaut had established, which seem not strictly accurate. One statement is this : assuming that Clairaut's fraction is '5? greater than ~ , then the density of the nucleus must be greater than the mean density of the Earth, but the ellipticity less than 5? 5 9 - . If Clairaut's fraction is greater than -* , the ellipticity must 4 4 be less than ^ , by Art. 336. But the statement that the density of the nucleus must be greater than the mean density of the Earth does not seem justified by anything in Clairaut : the nearest approach to it is in the second criticism of Art. 325, but this obviously falls short of the statement. BOSCOVICH. 319 Again Boscovich proceeds thus : Invenit autera ejusmodi fractionem majorem revera esse, et affirmavit ellipticitatem minorem erui e gradimm mensura ; unde intulit, ea duo conciliari non posse, nisi assumatur certa nuclei ipsius ellipticitas. The word minorem, which I have put in Italics, must be a misprint for majorem ; see Art. 349. Then for all that follows intulit there seems not sufficient authority ; the criticism in the first paragraph of Art. 325 is somewhat short of this. 486. Boscovich considers that more observations of pendu- lums and more measurements of degrees are required ; he admits that this would involve great labour and expense, but he adds, "at nihil est, quod Astronomorum patientia, et munificentia Regum superare non possit:" see his pages 511 and 512. Since his time the endurance of Astronomers and the liberality of Sovereigns have been largely exercised in the subject. He repeats on his page 513 that the fact of the compression at the poles might be admitted ; but the amount of the compression, and the true Figure of the Earth, were still quite uncertain. He finishes by giving on his pages 514... 516 approximate solutions of the problem to determine the Figure of the Earth, assumed to be an oblatum, from two measured degrees, one or both of which might be of longitude. 487. In forming an estimate of the treatise we must remem- ber that the author had prescribed to himself the condition of supplying geometrical investigations; so the Differential Calculus was not to be introduced. We must consider the treatise rather as the work of a professor for the purposes of instruction, than of an investigator for the advancement of science ; and then we may award the praise that the task proposed is fairly accomplished. It would have been more desirable to study Clairaut's work than to be confined to Boscovich's geometrical methods : but the ex- perience of our own university shews us that it is possible to find the methods used for teaching occasionally some years in arrear of those used for investigation. Although the mathematical processes seem a little out of date, yet Boscovich's treatise reveals, I think, great knowledge and judgment in Natural Philosophy. 320 BOSCOVICH. 488. Boscovich has an unpleasant habit of giving hints as to matters which will be found in other parts of his book, without supplying exact references ; I have observed many passages of this kind, and have not always been able to determine with certainty to what he is pointing. Thus we have on page 392, "de qua fortasse aliquid alibi infra;" on page 413, "videbimus;" on page 448, "porro videbimus;" on page 455, "ut infra patebit;" on page 466, "ut infra videbimus;" on page 506, "ut vidimus;" on page 507, " ut innui etiam;" on page 508, "supra innui." None of these allusions, however, are to matters of great import- ance ; but there is a passage of more interest on page 386 : Expediam autem, quod ad earn gravitatis legem pertinet, sive Tellus homogenea sit, in quo argumento felicissime sane Mac Laurirms se gessit, sive diversam in diversis distantiis deusitatem habeat, de quo casu multo aliter ego quidem sentio, quam summi etiam nostra? setatis viri senserint, quorum calculos laborare omnino censeo, cum Geometria duce ad conclusiones delabar prorsus contrarias eorum conclusionibus. I cannot see anything in the treatise which corresponds to "de quo... conclusionibus." Boscovich seems to dissent from only one person, namely Daniel Bernoulli, and D'Alembert had pre- viously objected to the same thing : see Art. 469. 489. Boscovich himself gave an abstract of his treatise in the Bologna Commentarii, Vol. IV. 1757, pages 353... 396. This supplies nothing of importance to our subject except three sepa- rate sentences, which I quote, because I do not understand them. With reference to the arc in Lapland, Boscovich says on page 389: et in Lapponia, adhibita huic postremo ilia correctione, quse adhibita est etiam a Bouguerio, et prater quam alias adliibendas non esse, ut ut ab alio nuper adhibitas, demonstrari facile potest. Bouguer's correction is that for refraction ; I do not know what the other corrections are, nor by whom they were proposed. After drawing an inference from Clairaut's theorem, Boscovich says on page 392 : quod ipsum cu*n ego in eo opusculo diserte affirmaverim, et Clerautii theorema ipsum ex mea theoria deduxerim, ipso Clerautio nominate, miratus sane sum in opusculo nuper in Hetruria edito, me STAY. 321 contri Cleruutium hac ipsa in re adduci testem pro homo^eneitate^t hoc ipsum meum indigitari opusculum. I do not know to what book Boscovich here alludes. Boscovich, as we shall see in our account of his commentary on Stay's poem, devised a curious method of treating discordant observations, so as to obtain the best result from them. It appears that he was now in possession of the method, and he makes a numerical application of it, though he does not give any expla- nation. He says on his page 392 : Invenio illud, quod in inemorato volumine nequaquam qusesiveram... I do not feel certain as to the meaning of these words, but I suppose the memoratum volumen to be his own treatise, of which he is giving an abstract : and then he seems to say that he had now solved the problem of the advantageous combination of obser- vations which had not been considered by him at the time of the publication of his treatise. The two serious misprints relating to the ellipticity, which occur on page 501 of the treatise, are reproduced on page 391 of the memoir : see Art. 482. 490. We now proceed to Stay's poem, to which Boscovich supplied a commentary. The title of the poem is, Philosophic Recentioris a Benedicto Stay . . .versibus traditce Libri X. cum adno- tationibus, et supplementis P. Rogerii Josephi Boscovich... This work consists of three octavo volumes, published at Rome, the first volume in 1755, the second in 1760, and the third in 1792. We have here a treatise on Natural Philosophy in Latin hexameters, extending to more than twenty-four thousand lines. Each volume contains copious notes ; and to the first and second volumes elaborate supplementary dissertations are added : these are all by Boscovich. The long interval between the publication of the second and third volumes was caused by the journeyings * and incessant occupations of Boscovich, which hindered him from completing his share of the work ; and he died before he had drawn up the intended supplementary dissertations for the third volume. T. M. A. 21 322 STAY. The number of students interested both in Natural Philosophy and in Latin Verse could scarcely ever have been large ; and is probably less now than formerly. Cambridge, I hope, has never been destitute of men of such tastes, but it is curious that the University Library does not possess a complete copy of the famous work by Stay and Boscovich. 491. Dugald Stewart, in his well-known Dissertation, after speaking in the highest terms of Boscovich, says : Italy is certainly the only part of Europe where mathematicians and metaphysicians of the highest rank have produced such poetry as has proceeded from the pens of Boscovich and Stay. It is in this rare balance of imagination, and of the reasoning powers, that the perfection of the human intellect will be allowed to consist ; and of this balance a far greater number of instances may be quoted from Italy, (reckoning from Galileo downwards,) than in any other corner of the learned world. Works edited by Hamilton, Yol. I. page 424. If I might venture to give an opinion, founded on such por- tions of Stay's work as I have read, I should say that it is rather versification than poetry, displaying technical skill rather than imagination. The subject, however, was not very favourable to his genius ; and sometimes his lines contrast unfavourably with the simple but elegant notes of his commentator. Boscovich, how- ever, had a high opinion of the text which he explained, for he speaks of it as operis sane immortalis ; see the De Litteraria Expeditions , page 390 : the French translation reduces this to outrage digne de rimmortalite. 492. The work is furnished with a preface by Boscovich, and with a letter to Benedict Stay from his brother Christopher Stay. The letter refers to Bacon and to Newton; see page xxix. While Newton's devout character is praised, the wish is gently expressed that he had known religion in its purity as well as its power. The part of the poem which concerns us consists of the latter half of the fourth Book and the former half of the fifth Book. We may say in general terms that we have an account of the results obtained by theory as to Attractions and the Figure of the Earth, and also of the operations carried on for measuring the dimensions of the Earth. STAY. 323 493. It may be satisfactory to the reader to have some specimens of Stay's verses. A passage in Book IV. beginning with line 1038 is interesting. Stay illustrates the fact that although attraction is exerted by every particle of matter, yet the disturbing effect of mountains or great buildings on a falling body vanishes in comparison with the downward action of the whole earth ; he finishes thus: Inter saxa quidem, glebasque, herbasque virentes Mutua vis hsec est, et Kgna, et dura metalla ; Tellus tota tamen longe, longeque trahendo Prsevalet, absorbetque leves has undique vires Ingens, atque illos conatus prsepedit omnes, Ut Sol, cum radios Ceelo jaculatur ab alto, extincta licet stellarum lumina velat. I will take next a passage beginning at line 1941 of Book IV. ; Stay has explained Newton's method of determining the Figure of the Earth, and then he proceeds to shew where it was defective, and to state that Maclaurin supplied the defect. At reperire suo num motu Terra diurno Illam debuerit, quam coni segmina priina Proscissi dant, induere, et circumdare formam, -<33que etiam si densa, fluensque fuisset, ut unda, In elite Vir, porro non hoc accepimus a te Inter munera magna, quibus nos undique ditas ; Fors voluisti, alii ut quid tantis addere possent ; Sic alios Rex ssepe suis ditescere gaudet Thesauris, atque in vulgus diffundere dona, Postquam ipse iuimensara fuerit largitus opum vim. Hoc donum, Laurine, tuum est; stupuere docentem Multa Caledoniis Mortales te quoque in oris. Inter multa tamen longe hoc prsestantius unum est : Illam nempe doces formam a Tellure fuisse, Gyros agglomerat dum circa se, subeundam, Si liquida, et molem foret seque densa per omnem, Atque, polos inter, medias attollier oras Mensura circum, dixi qua nuper, eadem Propterea debere, atque hinc quoque crescere eodem Ordine, quo dixi, paulatim pondera rerum, Inque polos illas gravitati accedere vires. 212 324? STAY. As another specimen I will take a passage beginning at line 712 of Book v.; it is part of the description of the operations of Maupertuis and his friends in Lapland : Postquam flumineo mensura est coguita dorso Ilia prior \ montes turn qua ratione adeundi "? TJndique prseruptis silvse stant inontibus altse Yerbera ventorum tan turn frangentia ramos Perpessse, nuiiquam flammas, diramque bipennein, Obstructs nivibus, mortali fors pede nunquam Tentatse ; jam sunt nudanda cacumina, Cseloque Illse ostentandse rupes, jam mentis ad iinani Radicem aerii, Kittim dixere Coloni, Hserendum est ; illic fabricanda patentia sursum Pastorum de more mapalia, suspicerentur Unde faces Cseli, et sublimes verticis ulnae, Et suiit multa locis aptanda, movendaque multis Instrumenta gravi molimine, Daedalus ille Praesertim multa quse fecemt arte Britannus, TJranie cujus tantum est munita labore \ Ipse gradus, graduumque dedit cognosse per arcuin Particulas seuas decies in quolibet uno, Atque harum totidem quoque fragmina particularum, Quse non, convexis nisi vitris, cernere, tantum est. Nimirum, genus hoc, arete conclusa supellex, Ne quid in offensu vario, com page soluta, Turbaretur, eos montes, prseruptaque cnrru, Sive levi potius scandebat culmina cimba, Consimilis cervo quam bellua juncta trahebat, Ocyor at multo, multoque ferocior illo, Perque nives, glaciemque, per horrida saxa volabat. Indigense, rude vulgus, iiiers, nullisque juvare Consiliis, operisque potens, cum ssepe viderent Circum alienigenas fundi, atque, ut sacra ferentes, Lente onus id vectare Viros, intus latitare Numina credebant, Divum et procedere magnam Matrem inter Gallos; namque illos stulta premebat Relligio, exanimesque Deos, et inania signa Thure coli, votisque jubens, et sanguine fuso. BOSCOVICH. 325 494. The supplementary dissertations with which we are con- cerned extend from page 359 to page 426 of the second volume. 495. The first dissertation is entitled De incequalitale gravi- tatis per superficiem telluris, et figura ipsius telluris ex cequilibrio: it occupies pages 3 5 9... 380. This may be described as an abridgement of the matter on the same topics given by Boscovich in the treatise we have already examined. Boscovich says on his page 3<U, referring to the former treatise : Ego rem totam ad solius finitse Geometrise vires redegi in memorato opuscule, Siugula fuse persequi, et accurate demonstrare non sinit ipsa horum supplementorum brevitas; quamobrem indicabo tanturamodo methodum, qnam adhibui, et theoremata praecipua, ac formulas inde erutas j ubi tamen occurrent qusedara et perpolita magis, et promota ulterius, quam ibi. I shall notice some miscellaneous matters of interest which present themselves. 496. In his Article 203, on page 359, Boscovich asserts more positively than in the former treatise, that a mass of fluid in equi- librium under no external forces must take a spherical form. 497. In his Article 209, on page 361, he is speaking about the deduction of the Figure of the Earth from the theory of gravity, and he says, "in qua perquisitione Newtonus incassum laboravit,... feliciter autem rem confecit Mac-Laurinus." This seems scarcely just to Newton, whose investigation was satisfactory as far as it went ; and this is admitted by Boscovich himself elsewhere ; while we do not know that Newton tried to do more and failed, as is sug- gested by the words incassum laboravit. See Art. 501. 498. In his Articles 228 and 229, on pages 366 and 367, we have a more elaborate investigation than in the corresponding part of the former treatise, which' we have noticed in Art. 468. He is discussing the case in which there is a spherical nucleus surrounded by fluid ; and in the present investigation, the radius of the nucleus is not assumed at first to be approximately equal to the radius of the outer surface of the fluid. In his Article 232, on page 368, he proposes the name f radio gravitatis, for what we have called Glair auCs fraction: see Art. 336. 326 BOSCOVICH. By the aid of what he had given in his Articles 228 and 229, Boscovich is now able to supply an investigation of Clairaut's theorem, which is rather more general than that in the former treatise : see his Article 237, on page 369. 499. His Article 238, on page 370, is important. He quotes the words from the second edition of Newton's Principia to which we have drawn attention in Art. 30, namely, " Ifec ita...adhuc major." It would however have been right to remark that the words were omitted in the third edition of the Principia. Boscovich adopts the same opinion as Clairaut, with respect to the origin of Newton's error ; but states it I think more clearly ; see Art. 37. Boscovich says : et hunc quidem Newtoni errorera Clerautius deprehendit, ac protulit. Censuit fortasse Newtonus conjectura quadam usus, et re ad geometricam tnitinam uequaqnam redacta, in quavis hypothesi, lit in casu homogeneitatis, vires in sequatore, et in polo, esse reciprocas distan- tiis, quas vidit magis augeri in polo, si inassa nuclei fiat major, ob excessum gravitatis in illam massam adjectam pro loco viciniore ipsi in polo. 500. His Article 244, on pages 371 and 372, is important. He is correct as to a matter in which there is at least the appear- ance of error in the former treatise : see Art. 466. At the end of his Article, Boscovich indicates that he is about to investigate a certain theorem more generally than in his former treatise : the theorem is that the increase of gravity in proceeding from the equator to the poles varies as the square of the sine of the latitude. On his pages 375 and 376, he gives tabular results as to the value of gravity at different places which are fuller than in the corresponding part of the former treatise, namely pages 479 and 480. 501. On his page 378, Boscovich expounds Newton's method of determining the Figure of the Earth ; he says in his Article 264 : Clerautius in opere de figura Telluris miratur, Newtonum vidisse figuram Telluri debitam hac methodo, velut trans nebulam quandam ; at mini quidem videntur prona omnia in hac ejus methodo Nihil in toto hoc progressu mini videtur alienum a sagaci quidem, sed et solida, et usitata Newtoni perquirendi ratione. BOSCOVICH. 327 But I do not find any such remark made by Clairaut as is here attributed to him ; perhaps Boscovich was really thinking of a sentence with respect to Newton, which occurs in the Essay on the Tides, by Daniel Bernoulli, Chapter II., Article vin. : Quant a son raisonnement, il n'y a peut-etre que lui, qui put y voir clair ; car ce grand homme voyoit a travers d'un voile, ce qu'un autre ne distingue qu'a peine avec un microscope. 502. The next dissertation is entitled " De deviationibus pendulorum ex asperitate superficiei terrestris, et methodo definiendi massam terrce : it occupies pages 380... 384 503. On his page 381, Boscovich refers to a figure which is not to be found in the book ; so the reader must draw it for himself. In the section we are now considering, Boscovich advocates the plan for determining the mass of the Earth which he had proposed in the former treatise : see Art. 476. He also suggests a modifi- cation of it. He would have constructed at royal expense in certain valleys immense reservoirs, so that they could be filled with water by the mountain streams, and again emptied at plea- sure ; then the position of an adjacent pendulum is to be observed before and after the reservoir was filled with water. As the form and dimensions of the reservoir would be exactly known the de- viation which the mass of water would produce in- the pendulum could be calculated, assuming the ratio of the density of the water to the mean density of the Earth : and then by comparison with observation this ratio would be determined. Boscovich manifestly held very decided opinions as to the duty of governments in encouraging science. 504. The next dissertation is entitled De veterum conatibus pro magnitudine terrce determinanda : it occupies pages 385... 389. Boscovich refers to a separate dissertation which he had pub- lished entitled, De Veterum argamentis pro Telluris sphcericitate : this I have not seen. The principal matter to notice here, is the detail of an investi- gation to which we alluded in Art. 475 ; he admits that there was a slight error in the result he formerly gave : his method is sound but laborious. 328 BOSCOV1CH. By comparing his result with that which I obtained in Art. 475, the following formula is deduced. 7T_1_J L J_ J>_ 7 2 6 80 336 46OS - that is - Wj u 2 -u s -u 4 ... = 2 log 2, This may also be established thus : We have |" {sin- J + 5^fi J - ? J c?r = 2 log 2 - f ; for the indefinite integral of the expression under the integral from which the definite integral follows. Again ^l+V^. so that the definite integral = f [2 f 1 M 7 *" 3 ^ _ a A L ^o I ^ * Integrate with respect to r first ; thus we obtain Thus the required formula is established. 505. The next dissertation is entitled De primis recentiorum conatibus pro determinanda magnitudine telluris: it occupies pages 390... 393. BOSCOVICH. 329 In his Article 304 on page 391, after shewing that a certain process which seems theoretically advantageous fails by reason of practical difficulties, he Concludes with this reflection : ut qiue methodi directed videntur primo fonte omnium aptissimse ob theorise simplicitat-em, plerumque fato quodam couditionis humanse fiant maxime omnium ineptse, et per ambages ssepe indirectas segre" demum eo, quo tenditur, liceat evadere. In his Article 307 on page 3.92, he points out the changes successively made in the French degree of the meridian originally measured by Picard, and concludes with this reflection : Inde autem vel in hoc solo Piccarti gradu facile constat, per quas ambages, et inter quos errorum scopulos ad veritatem emergat humana mens. 506. The next dissertation is entitled De dimensione graduum meridiani, et paralleli : it occupies pages 393... 400. This gives a good sketch of the process of measuring an arc of meridian or of longitude. 507. The next dissertation is entitled De figura, et magni- tudine terrce ex plurium graduum comparatione : it occupies pages 400... 405. In his Article 337 on page 402, Boscovich works out one case to which he had only alluded on page 490 of his former treatise ; namely having given the length of a degree of meridian at one latitude and the length of a degree of longitude at another, to determine the axes of the Earth. But he seems to attach the greatest importance to some approximate formula? for the length of a degree of meridian or of longitude to which he had drawn attention in the last two pages of his former treatise. These formula? all depend on the following approximate expression for the radius of curvature at any point of an ellipse, -^ cos 2 X, where X is the latitude, and a, b, e have their usual meaning. He says as to his formulae in his Article 344, on page 403 : Ego quidem vix crediderim posse simpliciore, et magis uniform! metliodo solvi haec quatuor problemata. . . . 330 BOSCOVICH. 508. The next dissertation is entitled De recentissimis gra- duum dimensionibus, et figura, ac magnitadine terrce inde deri- vanda : it occupies pages 406... 426. Boscovich takes the same five arcs as in his former treatise ; see Art. 481. These furnish as before ten binary combinations, and therefore ten values of the ellipticity : see Art. 482. He gives the result in a Table on page 408, which may be compared with that on page 501 of the former work. He has used a slightly different formula for computing the ellipticity, so that in the later Table each denominator should exceed by 2 the corresponding denominator in the former Table. The ellipticities deduced from the ninth and tenth combinations are however quite wrong in the later Table. 509. Boscovich lays great stress on the discrepancies between the various measures of degrees ; he attributes them mainly to deviations of the pendulum, produced by inequalities in the sur- face and the crust of the Earth. He in fact holds, as in his former treatise, that very little was really known as to the true figure of the Earth : see Art. 484. He expresses his opinions with some em- phasis, and indeed it seems to me that he has allowed his feelings to disturb his attention or his judgment, for there are various misprints and some difficulties in the dissertation. In his Article 360, on pages 409 and 410, he alludes to Bouguer's hypothesis, that the increment of the length of a degre.e of the meridian in passing from the equator to the pole varies as the fourth power of the sine of the latitude ; but he has omitted Bouguer's name, so that the hypothesis seems to be ascribed to Clairaut or Maupertuis. In his Article 365, on page 411, he refers to an objection he had formerly expressed when Maupertuis was supposed to have settled the exact Figure of the Earth ; and for this he says, " tanquam audacissimus, et ineptus trad net us sum." He goes on to speak of " illam ipsam tantam compressionem, quam in eo opusculo Maupertuisius vulgaverat," . . . ; but this is not accurate, for Maupertuis did not explicitly assign any value to the compres- sion in his book, though he gave the length of his own degree, and also what he then considered to be the correct length of BOSCOVICH. 331 Picard's degree: see Maupertuis's Figure de la Terre...p2igQ 126. But we have seen in Art. 177, that Clairaut once suggested inci- dentally a very large value of the ellipticity as obtained from the operations at the polar circle. In his Article 371, on page 413, Boscovich says : Multo est major utique hsec ipsa Telluris asperitas, utut tam exigua respectu totius diametri, et multis partibus major, quam, quae totam etiam possit quadringentarum hexapedarum insequalitatem parere, quam inter Quitensem, et Laponicum gradum observationes exhibent ; . . . This is not intelligible. The difference between the lengths of a degree of the meridian in Lapland and Peru is according to Boscovich's own Table 671 toises, not 400. But perhaps by ince- qualitas he means not the difference of the two lengths, but the deviation from some theoretical standard : if so, he should have explained what the standard was, and how the deviation was estimated. 510. On his page 414, Boscovich criticises some statements made by Maupertuis in a work on Geography ; and on his page 416 he animadverts on the Article relating to the Figure of the Earth in the Encyclopedic : the objections amount to this, that sufficient attention was not paid to the irregularity of the Earth's surface and crust. Boscovich gives us on his page 416 the following depressing view of the course of human investigations : At et hie quidem notare, et admirari licet humanae gentis condi- tionem ubique uniformem, quse per crebras positiones falsas, erroresque atque errorum correctiones multiplices, post erronetts observationes, erroneas etiam ratiocinationes multas segre demum per longam observa- tionum, et contrariarum opinionum seriem enitatur ad veritatem. 511. The most important part of this dissertation is that contained in pages 420... 425. Boscovich here explains a method of his own invention for combining discordant observations so as to evolve an advantageous result. As applied to the present subject it may be stated thus : to determine the generating ellipse of the Earth's surface from the measured lengths of de- grees of the meridian, under the two conditions that the sum 332 BOSCOVICH. of the negative errors shall be numerically equal to the sum of the positive errors, and that each sum shall have the least possible value. Boscovich's exposition of his method takes a geometri- cal form : it is simple, clear, and instructive. Laplace gave Boscovich's method, divested of its geometrical form, in the Paris Memoires for 1789 ; and subsequently in the Mecanique Celeste, Livre in. 40. Boscovich exemplifies his method by applying it to the five arcs he had adopted; see Arts. 481 and 508 : these furnish ^= for the ellipticity. The residual errors for the length of a degree in toises for the Equator, Cape of Good Hope, Italy, France, and Lapland, are respectively 0, - 79 '2, 93*8, 75*9, and 90'o. In the French translation of Boscovich's former treatise, besides this example another is given, which in- volves nine measured arcs. 512. The poem of Stay, with the commentary of Boscovich, constitute a good elementary exposition of the principal results which had been obtained relative to our subject. It may be doubted whether the system on which the book is constructed is the most economical of the student's attention ; for in fact various points are often treated three times, first in the verses, next in the notes, and finally in the supplementary dissertations. But probably some readers, for whom the dissertations would be too elaborate, might find the more popular parts of the work entertaining and instructive. 513. It will be convenient to notice here, though a little out of date, the French translation of the De Litteraria Expeditione of Maire and Boscovich : this was published at Paris in 1770 under the title of Voyage Astronomique et Gfographique, dans I'Etat de VEglise.... This is in quarto, containing a Title and Introductory matter on xvi pages, and the text on 526 pages ; there are also four Plates and a Map. Some notes are added to the translation, and also a copious Index : the map, notes, and index render the translation more useful than the original. The name of the translator is not given ; but in the life of Boscovich in the Bio- graphic Universelle, the De Litteraria Expeditione... is said to be BOSCOVICH. 333 "traduit en franc,ais, sous le nom de I'abbe' dmtelain, par le P. Hugon, jesuite." See also La Lande's Bibliographic Astrono- mique, page 515. 514. In the part of the translation with which we are con- cerned there are some matters which may be^briefly noticed. On pages 449... 453 there is a long note of a controversial character relating to D'Alembert ; we shall mention it hereafter in connexion with D'Alembert's Opuscules Mathitmatiques, Vol. VI. On pages 478... 483 there are notes giving the results ob- tained by measurements in Hungary, Piedmont, and North America, which had been executed since the publication of the original work. On pages 501... 512 we have an important note. This gives us first an account of Boscovich's method of treating discordant observations, which is a translation of the exposition by Boscovich himself, published in his commentary on Stay's poem: see Art. 511. Then the method is also applied to the case of nine measured arcs, namely, the five formerly taken by Boscovich, together with four others. Also some remarks are made as to the density of a supposed spherical nucleus in the Earth. A curious note occurs on Article 11 of Boscovich's treatise. Boscovich is speaking of relative motion, and he says that if the space in which the Earth is situated has a motion equal and opposite to that of the Earth, then the Earth itself is at rest; the note then adds: Voici de quoi rassurer ceux qui apprehendent que le double mouve- ment de la terre, dans les systemes de Copernic et de Newton, ne soit oppose an sens litteral de PEcriture sainte. Rien ne les empeche de supposer la terre immobile, sans rien de"ranger a 1'economie de ces systemes. A note on pages 36 and 37 of the translation informs us that various measurements of degrees were undertaken at the sug- gestion of Boscovich ; namely, those in Austria and Hungary by Liesganig, that in Piedmont by Beccaria, and that in North America by Mason and Dixon. The connexion of Boscovich with the last is thus stated : 334 BOSCOVICH. Enfin dans son voyage en Angleterre, il a represente a la Societ6 Royale 1'avantage qu'il y auroit de faire mesurer un degre en Amerique, avec d'autant plus a raison, que depuis que 1' Astronomic est perfec- tionnee, 1' Angleterre n'avoit rien fait pour connoitre la figure de la Terre. The operations in England, in India, and at the Cape of Good Hope, since the time of Boscovich, have removed the reproach which is here cast on us. Perhaps we may hereafter have mea- surements made in Canada, Australia, and New Zealand. A note on page 15 records the name of a person who corrected the error of Keill and Cassini ; see Arts. 76 and 81 : M. des Roubais, Inge*nieur charge de poser les signaux, donna dans un Journal de Hollande, la demonstration, que les degres decroissans vers le pole, faisoient la terre allongee. See La Lande's Bibliographic Astronomique, page 372. CHAPTER XV. MISCELLANEOUS INVESTIGATIONS BETWEEN THE YEARS 1741 AND 1760. 515. THE present Chapter will contain an account of various miscellaneous investigations between the years 1741 and 1760. I shall not in future record the titles of memoirs relating to observations of pendulums ; as those which present themselves after the period at which we have arrived are given in well-known works. See La Lande's Astronomic, third edition, Vol. III. pages 43 and 44; Reuss's Repertorium...Vol. v. pages 79 and 80 ; and the Article on the Figure of the Earth, in the Encyclopedia Metropolitans 516. A work was published at London, in 1741, entitled Mercators sailing, applied to the true figure of the Earth. With an introduction concerning the discovery and determination of that figure. By Patrick Murdoch, M.A., Hector of Stradishall, in Suffolk. This is a quarto volume, containing xxxii + 38 pages, and three plates of figures, The title points out that the work consists of two parts ; we are principally concerned with the first part : on this a few remarks may be made. 517. The most distinctive part of the book is the treatment of the hypothesis that the Earth is not homogeneous, but has a central nucleus denser than the surrounding fluid. Murdoch maintains that if this central nucleus is spherical, the ellipticity of the external fluid surface will be less than on the homogeneous 338 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. hypothesis ; but if the central nucleus is an oblatum similar to the external fluid surface the ellipticity will be greater than on the homogeneous hypothesis. To shew this, he first gives some gene- ral reasoning on his page xxi ; then he briefly sketches a mathe- matical investigation, and states the formula to which it leads on his pages xxii and xxiii; and from his formulae he deduces nu- merical results on his page xxiv. But this distinctive part of the book is unsatisfactory. In the first place, no attempt is made to shew that the mass is in relative equilibrium ; but assuming it to be in that state, an equation is obtained by considering equatorial and polar columns. In the next place, since there is supposed to be a hard nucleus the columns cannot be produced to meet at the centre, and so Murdoch has to make an arbitrary supposition. This supposition expressed in modern language is that the pressure of the fluid on the nucleus is the same at the points where the equatorial and polar columns meet the nucleus. Since his results are based on these unsatisfactory principles, they cannot be accepted. 1 have, however, verified his formulas, and find that on his assumptions they are correct. I have not gone over the calcula- tions by which his numerical results on his page xxiv are obtained. 518. Let us take one example of his numerical results from another place, namely his page xxvi. He says : ...For in one of these Examples, where the redundant Matter was a Sphere with the Radius j of the Semidiameter of the Equator, if we no compute its accelerating Force at the Pole, we shall find it about of the whole ; and consequently the whole Density of the concentric Sphere would be to that of the ambient Matter as 42 to 1. Proportions which will not, I presume, be thought very natural; whereas, if the redundant Mass is a Spheroid similar to the Earth, their like Diameters 4Q being as 1 and 4, its accelerating Force at the Pole will be only , and the whole Density of the Spheroid to that of the ambient Matter, in little more than the Eatio of 1307 to 1000. MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 337 On this passage, I remark that the great discrepancy between the two results, when so slight a change is made in the hypothesis as the transition from a spherical nucleus to an oblate nucleus, should have shaken Murdoch's faith in his whole process. But at tlie same time I do not see how his numerical results are obtained. Let er denote the density of the redundant matter, and p the density of the ambient matter. Let a denote the major semiaxis of the Earth, b the minor semiaxis, and e the ellipticity. Then oo assuming the correctness of his fraction -^ , which depends on his preceding formulae, we have - 38 so that 5-3 -^j a 1 4.* a- _ 38 x 64 _5 38 x 64 p ~ 62 1-f 3e ~ 62 If we put e = 0, we get - less than 40; if we put, as Murdoch does elsewhere implicitly e=^j, we get - less than 39. I presume that he means to say we get - - = 42. In the second example proceeding in a similar way, I find approximately 4 <r_ 48x64 * 5 48 x 64 p~ 952 l + 2e~ 952 so that - is about 3, and about 4 : this differs altogether P P from Murdoch's result. 519. On page xxvi. the passage is quoted from the second edition of Newton's Principia which corresponds to that from the first edition which we have quoted in Art. 37. T. M. A. 22 338 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. Murdoch considers Newton's language to indicate that he in- tended his nucleus to "be not spherical, but oblate ; and Murdoch thinks that D. Gregory in his Prop. 52, Lib. HI. overlooked this. But I do not believe that Newton really intended to discriminate between these two forms for a nucleus. On page xxvii. there is a reference to the Principia, Lib. I. Prop. 91, Cor. 2 ; but this passage has no bearing on the matter which Murdoch is discussing. On page xxix. there is a note on the erroneous notion which Cassini held as to the figure of the Earth, in these words : " He has, I am told, of late ingenuously owned his Mistake." On his page xxxi. Murdoch is speaking of the operations of Maupertuis, and exhibits that inaccuracy which by some fatality seems to cling to all the derived accounts of this measurement : see Art. 199. Murdoch says "... after proper Allowances for the Refraction of Light, the Precession of the Equinoxes and Mr Bradley 's Equation...". But in fact no allowance was made for refraction, as Murdoch himself admits on his page xxx. By Mr. Bradley's equation is meant what we call Aberration. Besides Precession and Aberration there was a correction for Nutation. 520. The part of Murdoch's work which is called Mercator's sailing applied to the true figure of the Earth, does not really fall within our scope; and so we shall not give any great attention to it. We may say generally, that the object is to construct maps of the Earth's surface, assuming the form to be an oblatum, like the maps on what is called Mercators projection for a spherical form : or it is practically equivalent to this. Murdoch is unfortunate in the value he adopts for the ellip- 02 ticity ; in modern notation he takes e 2 = '022, so that ^ which is approximately the ellipticity, is about ^-: . I presume that he deduced this value from the degree in Lapland, combined with Picard's degree, taking the latter at the amount assigned to it MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 339 by Maupertuis in his Figure de la Terre...p&ge 126. This amount for Picard's degree was soon afterwards found to be too small. In consequence of the very large value assigned to the ellipticity, maps constructed according to Murdoch's tables would in general be more erroneous than maps constructed on the hypothesis of the spherical form of the Earth. 521. Murdoch's work was translated into French under the title Nouvelles Tables Loxodromiques. . .par M. Murdoch. Traduit de V Anglais. Par M. De Bre'mond... Paris, 1742. This is an octavo volume consisting of xvi + 158 pages, be- sides the Privilege du Roy on four pages. There are four plates of figures. r J he translator dedicates the book to Le Comte de Maurepas, the French minister who was very much concerned with the expeditions sent to Peru and to Lapland. The following sentence from page vii. is of interest : Malgre ce qu'un autre Auteur Anglois pretend qu'a pense Strabon sur Fapplatissement de la Terre, celui-ci a 1'equite d'avouer que tous les Pliilosophes et les Geograplies n'attribuoient point a la Terre d'autre Figure que celle d'un Globe parfait, avant la fameuse Experience faite SL Cayenne en 1672. par M. Eicher Astronome Frangois. On page 19, there is a note on the passage of Strabo; and it is maintained that the passage does not shew Polybius to have been acquainted with the true figure of the Earth. It ought to have been stated that this note is due to the translator, and no*t to Murdoch himself. I have noticed the passage in Strabo already : see Art. 152. 522. The pages 27... 46 consist of an important addition sent by the author to the translator. The essence of this addition is to be found on page 43, namely formulae which give the attraction at the pole and at the equator, both for an oblatum, and an oblongum. These formulae are not demonstrated, but differential expressions are investigated which will lead to the formulae by integration. The formulas are correct. 222 340 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. Let M denote the ratio of the attraction at the equator of an ellipsoid of revolution to the attraction of a concentric sphere touching the ellipsoid at that point ; let N denote the ratio of the attraction at the pole of the ellipsoid of revolution to the attrac- tion of a concentric sphere touching the ellipsoid at that point. Then whether the ellipsoid of revolution be an oblatum or an oblongum, we shall have tf 3 2+ M =2' This formula is given, though not quite with this notation, on page 44 : by attending to the formula we can discover the meaning of the first seven lines of page 45, to which the printer has not done justice. 523. The addition which Murdoch sent to his translator appeared in the same year as Maclaurin's Fluxions: but, as we have seen, Maclaurin had been substantially in possession of the results respecting the attraction of an ellipsoid of revolution at the time he wrote his Prize Essay on the Tides. Thus Maclaurin's claim to be the first who completely solved the problem of the attraction of an ellipsoid of revolution on a particle within the body, or on its surface, remains untouched. 524. Another addition sent by the author to the translator is given on pages 104... 108. This relates to the part of the work which treats on the construction of maps. The addition is due to a suggestion made by Maclaurin to Murdoch, and it effects a great improvement in the mathematical investigation. See Maclaurin's Fluxions, Arts. 895... 899. 525. The translation is not very well executed. Some pas- sages are unintelligible, where the original is quite clear ; as an example may be mentioned a passage about Antipodes, on page 129 of the translation and page 18 of the original. The following is a curious specimen of a misprint. On page 142 of the translation, we have &E1. 4. By turning to the original, page 29, we find it should be 6 El. 4; here El. stands for Elements, and so what is meant is, Eudid> VI. 4. MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 526. In the Paris Mtmoires for 1742, published in 1745, there is an article, entitled Sur la Figure de la Terre, which occupies pages 86... 104 of the historical part of the volume : the article is by Mairan, as appears from page 92. We have here a notice of Clairaut's Figure de la Terre, pre- ceded by a sketch of the history of the subject; there is, however, nothing of importance for us. It is remarked that even before the observations made by Richer at Cayenne, on the length of the seconds pendulum, it had been suspected at the Academy that the length ought to become shorter as the equator is approached ; to support this remark the fourth Article of Picard's work is cited, Mairan uses the word Pesanteur, not in the sense adopted by Maupertuis and Clairaut, but for the Earth's action apart from centrifugal force : see Art. 299. Mairan also uses the words gravitation, gravite, attraction; but without any apparent aim at precision : see his pages 98 and 103, 527. The results obtained in the geodetical operations which had been carried on during some years in France, were published at Paris in 1744, by Cassini de Thury under the title of La Meridienne de I' Observatoire Eoyal de Paris, vfrifiee dans toute Vet endue du Eoyaume par de nouvelles observations. The volume is in quarto : it contains Half-title, Title, Table, then pages 292 + ccxxxv ; followed by an alphabetical list of places, and the Privilege du Roy : there are xiv Plates. On pages 4 2... 51 of the historical part of the Paris Memoires for 1744, published in 1748, we have an account of this work. On pages 237... 244 of the Paris Mtmoires for 1758, published in 1763, we have some corrections by La Caille of the results obtained in the work. The Discours Preliminaire with which the volume commences gives a brief account of the operations. The formal admission is made by Cassini, that the length of a degree of the meridian increases from the equator; and that the Earth is therefore oblate : see his page 25. Thus the error which he had maintained after his father and grandfather is abandoned. 342 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. The present work supersedes the De la Grandeur et de la Figure de la Terre, and has in its turn been superseded by the Base du Systeme Metrique... 528. The volume of the Paris M6moires for 1745, which was published in 1749, contains a controversy between Clairaut and Buffon, which we must notice. Clairaut, in investigating the Lunar Theory, obtained for the motion of the apse line a result about half as great as that assigned by observation. In order to explain the difficulty, he proposed to change the law of attraction, by adding another term to the ordinary expression, which varies inversely as the square of the distance. But he soon discovered and admitted his error as to the Lunar Theory : see page 577 of the volume. In the controversy, Buffon attempted to shew that it was necessarily impossible for the law of gravity to be expressed by the aggregate of two terms, one varying inversely as the square of the distance, and the other varying inversely as the fourth power of the distance : but his reasons are quite inconclusive. Clairaut maintained justly that there was nothing absurd in such a supposition. The controversy consists of six papers, three by each disputant ; but it does not seem that all which was spoken or written, was printed. Clairaut refers to the discrepancy between theory and obser- vation relative to the figure of the Earth, as throwing suspicion on the ordinary law of attraction; but he admits that he had not attempted to discuss the problem on his hypothetical law : see his pages 531 and 547. 529. We have next to notice a memoir entitled Eustachii Zanotti De Jtgura Telluris. This memoir was published in the De Bononiensi Scientiarum et Artium Institute atque Academia Commentarii, Vol. n. Part 2, Bononise, 1746. The memoir occupies pages 210... 227 of the volume. Assuming that the Earth is an ellipsoid of revolution, Zanottus shews how the dimensions of the ellipse may be found from the measured lengths of two arcs, either of the meridian, or of a normal section at right angles to the meridian. There is nothing MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AM) 1760. 343 remarkable about the geometrical processes. Zanottus employs the theorem which had been demonstrated by Clairaut respecting the radius of curvature of the section at right angles to the meridian, and he refers to Clairaut' s memoir: see Art. 161. There is an account of the memoir on pages 442... 451 of Part 1, of Vol. ii. of these Bologna Transactions, which is dated 1745. This is a very lively and interesting notice. The liber- ality of the French king is commended for undertaking the expense of the Arctic and Equatorial expeditions. Zanottus thought that it would be an honour to the Italians, if they con- tributed something towards the solution of the problem, before Godin returned from America, and finally settled the question. Accordingly, Zanottus proposed to execute a measure of an arc at right angles to the meridian of Bologna ; he explained and enforced his plan in a meeting of the Academy, but without success. We read ...Invitavit ; rogavit ; obsecravit. Multos etiam commovit ; laboris soeios sibi adjunxit ; sed Ludovicus Magnus in corona non adfuit. Ta- men, etsi rem noil perfecit, spem retinuifc, et voluisse non poenituit. Quod dicimus, ut qui italorum ingenio nihil tribuunt, voluntati certe, si quid voluntas apud ipsos mereri potest, dent aliquid. Quamquam et ingenio tribuent fortasse non nihil, si Cassinum meminerint fuisse nos- trum. 530. The volume of the Paris Htmoires for 1747, published in 1752, contains a memoir by La Condamine on an invariable measure of length. An abstract of this memoir in viii pages is found in some copies of the work XII of Art. 352. The volume of the Paris Mtmoires for 1748, published in 1752, contains a memoir by Cassini de Thury on the junction of the Meridian of Paris with that which had been traced by Snell in Holland : see Art. 105. 531. A problem occurs connected with our subject on pages 175, 176 of the Memoires de Mathe'matique...par divers Scavans... VoL I. Paris, 1750. The problem is entitled Supposant la loi ^attraction en raison inverse du quarre de la distance, frontier la 3i4 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. nature du solide de la plus grande attraction. Far M. de Saint- Jacques. The author's name is elsewhere increased by the addition of the words de Silvdbelle. The problem is well solved in two ways. In one solution, the early method of treating problems in the Calculus of Variations is used. In the other solution, a simpler method is adopted. Both ways of solution have been since reproduced. See my History... of the Calculus of Variations, pages 361 and 484. For some account of Saint-Jacques de Silvabelle, see De Zach r s work, U attraction des montages... page 588. 532. The Philosophical Transactions, Vol. XLVIII. part I. for the year 1753, published in 1754, contains an article on our subject under the following title : An account of a Book intitled, P. D. Pauli Frisii Mediolanensis, &c. Disquisitio mathematica in causam physicam figurce et magnitudinis Telluris nostra3 ; printed at Milan, in 1752. Inscribed to the Count de Sylva, and consisting of Ten Sheets and a half in Quarto : By Mr. J. Short, F.R.S. This article occupies pages 5... 17, of the volume. I have never seen this dissertation by Frisi ; but I presume, it was incorporated by Frisi in his Cosmographies . . .Pars altera. . . which was published in 1775, of which we shall give an account hereafter. Short speaks in high terms of Frisi, thus : This does not, however, in the le^st detract from the merit of F. Frisi ; who discovers throughout this work much acuteness and skill, joined with all the candour and ingenuity, that become a philosopher. And as he has not yet exceeded his 23d year, it may be expected, that the sciences will one day be greatly indebted to him ; especially as we find him actually engaged in composing a complete body of physico- mathematical learning. 533. The most important part of the article consists of the defence of a passage in Newton, which Frisi had misunderstood and asserted to be erroneous. The passage is that of which we give an account in Art. 22. Newton uses twice in the sentence MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 345 the phrase in eadem ratione. Then as Short says : " In which the expression eadem ratione occurring a second time has misled F. Frisi and others, to think this last ratio to be likewise that of the axes, or of 101 to 100." In fact, however, in the second case, the ratio is not that of 101 to 100 ; but that of 126 to 125J. The context shews clearly, that Newton is quite correct in what he means to say. Frisi, however, was not convinced that the error lay with himself, instead of with Newton : see page 123 of the book cited in Art. 532. As we have seen in Art. 137, Maupertuis also appears to have been misled by Newton's words. 534. The defence of Newton seems to have been supplied by Murdoch, as appears from the following passage in Short's article : I sent F. Frisi's book to my ingenious and learned friend the reverend Mr. Murdock, Fellow of this Society, who has fully cousider'd the question concerning the figure of the earth ; and who, after having perused the book, and discover'd the above mistake of F. Frisi, sent me the above theorem, and its demonstration. He likewise sent me the following theorems, which, he says, he had communicated to M. de Bremond, in the year 1740, when he was translating his treatise on sailing : But M. de Bremond dying soon after, those, who had the care of publifihyjg the translation, printed it incorrect in several places ; particularly the theorems for the prolate spheroid : On which account, he says, if they are thought worth preserving, they may be inserted in the Philosophical Transactions. Accordingly expressions follow, which amount to giving the values of the attraction of an oblatum or an oblongum, at the pole or at the equator. But it was unnecessary to publish them now, because Maclaurin had completely solved the problem of the attraction of an ellipsoid of revolution on a particle at the surface. Moreover all that is here given is also in my copy of Bremond's translation, pages 43 and 44, and printed quite correctly : so that the above statement seems unjustifiable : it is however possible that the original page was cancelled, and a reprint substituted. I may say that I do not understand how a numerical result is obtained, which is ascribed to Frisi on the fourth line of page 10. And on page 17, after "whose tangent is y(m* l) n some words follow which I do not understand, but which seem to me 346 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. unnecessary. Murdoch spells his name so himself; but others sometimes spell it Murdock. 535. We pass to another article which is connected with Short's, and is published in the same volume of the Philosophical Transactions. This is entitled A Translation and Explanation of some Articles of the Book intitled, Theorie de la Figure de la Terre ; by Mons. Clairaut, of the Royal Academy of Sciences at Paris, and F.R.S. This article occupies pages 73... 85 of the volume. I am not certain, but presume, that the paper is by Clairaut himself; it is written in the first person throughout, but it is not ascribed to Clairaut in Maty's General Index to the Philosophical Transactions. 536. Frisi considered that as to the Figure of the Earth, Boscovich underestimated the observations, while Clairaut, Bouguer, and others, underestimated the theory of Newton. Short, in his account of Frisi's dissertation, quoted the opinions. The present paper begins thus : Mr. Short, in his account of Father Frisius's Disquisitio mathematica in causam physicam Jigurae et magnitudinis telluris nostrae, having reported that philosopher's sentiments on my reflections upon the same matter, without taking the trouble to examine whether they were founded upon the truth or not, I find myself under the necessity of laying before the Royal Society the passages of my book, which, having been misunderstood by F. Frisiug, have occasioned the misconstruction, which he has made of my sentiments, either upon the trust I give to the actual operation made for discovering the figure of the earth, or Sir Isaac Newton's theoretical inquiries about the same subject. The expressions of Father Frisius, referr'd to by Mr. Short, are as follow : Quia tamen plerique omnes hucusque, aut nihil pro figura telluris determinanda ex iis observationibus deduci posse cum geometra celeber- rimo Ruggero Boscovik autumarunt, aut exinde cum 111. Clairaut, Bouguer, aliisque, contra incomparabilem virum ac prope divinum Isaacum Newton insurgentes, admirabilem ipsius theoriam facto minus respondenteni dixerunt, assignatamque in prop. 19. lib. 3. Princip. Mathem. terrestrium axium proportionem, a vera absonam omnino esse, MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 347 alios mihi observatiouibus parum, alios nimis tribuere visum est, omnes fernie oppositis erroribus peccasse, ubi res neque aurificis lance, neque molitoris, ut aiunt, statera librandse sunt. 537. Clairaut makes various conjectures as to what was the precise meaning of the charge brought against him by Frisi. Clairaut shews quite clearly that he had not given undue im- portance to observations, and had not undervalued Newton. The paper contains translations of the sections 51, 68, 69, and a por- tion of 70, from the second part of Clairaut's treatise. With respect to the matter we discussed in Art. 533, Clairaut says : After F. Frisius has examined himself the 19 problem of the third book of the Principia, the truth of which is incontestable, he finds, by his own mistake, a disagreement with the result of that proposition, and charges that illustrious author, without the least apology, with an error, which, says he, (quite from the purpose) is the sixth, that has been found in the same work, and also gives an enumeration of the five others, altho' they are not at all concerned in the question. 538. The volume of the Paris M^moires for 1751, published in 1755, contains a memoir by Bouguer, entitled Eemarques sur les observations de la parallaxe de la Lune, quon pourroit faire en meme temps en plusieurs endroits, avec la me'thode d'tfvaluer les changemens que cause a ces parallaxes la Figure de la Terre. The memoir occupies pages 64... 86 of the volume. There is an account of it on pages 152... 158 of the historical part of the volume. The memoir contains some interesting mathematical results, connected with the curve which Bouguer called the gravicentrique in his Figure de la Terre : see Art. 363, Bouguer maintains very sound opinions on the subject he discusses. If we were uncertain as to whether ,the figure of the Earth is oblate or oblong, then observations of the Moon's parallax might remove the uncertainty. But at the actual epoch this point was settled ; the only question was to fix the amount of the ellip- ticity of the oblate figure, and the observations could not, practically, be of sufficient accuracy for this purpose. But if we assume a value of the ellipticity, the corrections which have to be made to the 348 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. observations of the parallax in consequence of the figure of the Earth may be calculated : and the results will be important and useful. 539. La Caille was sent from France in 1750 to the Cape of Good Hope for the purpose of making astronomical observations. He proposed to determine the positions of the southern stars ; the parallax of the Moon, of Mars, and of Venus ; and the latitude and longitude of the Cape. La Caille resided in the country from April 1751 until March 1753. Besides the duties which he had specially undertaken, he found time to measure an arc of the meridian. The amplitude was rather more than 1 13' ; and he obtained 57037 toises for the length of the degree of the meridian which has its middle point in latitude 33 18'S. La Caille also determined the length of the seconds pendulum. The details of the operations connected with our subject are given in the Paris Mtmoires for 1751, published in 1755. The volume contains two memoirs embodying observations made by La Caille: the pages 425... 438 are devoted to the lengths of the degree and of the pendulum. There is also a short account of the voyage on pages 519... 536 of the volume. La Caille touched at Rio Janeiro on his outward passage, and there he met Godin who was returning from his long sojourn in South America. La Caille states on his page 524, that the southern hemisphere has more stars than the northern ; this statement is confirmed by actual enumeration : see Monthly Notices of the Royal A stronomical Society, Vol. xxxi. page 30. There is a notice of La Caille's voyage and work on pages 158... 169 of the historical portion of the Paris Me'moires for 1751. In one point this contradicts La Caille ; for it says that he deter- mined the position of Rio Janeiro, while he says himself that it was unnecessary for him to do this as he had been anticipated by Godin. The positions of the stars in the southern hemisphere, between the Pole and the Tropic of Capricorn, as determined by La Caille, are given in pages 539... 59 2 of the Paris Memoir es for 1752. MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 349 La Caille found it expedient to construct fourteen new constella- tions; but at the same time he suppressed the constellation of Charles s Oak, which he considers that Halley had fabricated by pillaging fine stars from the neighbourhood. He says on his page 591 : On n'y trouvera pas la constellation nouvelle que M. Halley a inseree dans son Planisphere en 1677, sous le nom de Robur Carolinum, parce qne j'ai rendu an Navire les belles etoiles que cet Astronome, age alors de vingt-un ans, en a detachees pour faire sa cour au roi d'Angleterre. Quelque louable qu'ait ete ce motif, je ne puis approuver la faQon dont M. Halley s'y est pris pour faire passer sa constellation ; . . . 540. A volume was published in Paris in 1763, entitled Journal historique du Voyage fait au Cap de Bonne- Esperance... 12mo., pages xxxvi + 380, besides the Approbation on four pages. There is a planisphere of the stars between the South Pole and the Tropic of Capricorn, which is reduced from that published in the Paris Mtmoires for 1752 ; and a map of the country in the vicinity of the Cape, which is reduced from that published in the Paris Me'm&ires for 1751 : the triangles of the survey are marked on the map. As to the authorship of the book see La Lande's Bibliographie Astronomique, page 482. On page 25 of this book we find the work La Meridienne de Paris verifie'e, ascribed to La Caille, though his name is not on the title-page. Delambre also considers that the entire operation belonged to La Caille : see the Base du Systeme Me'trique...Vo]. III. Avertissement, page 13. Among the memoirs ascribed to La Caille in the Journal historique..., on pages 71 and 72, we have one sur la precision de la mesure de M. Picard, and one sur la base de Ville-Juive : the titles, however, do not seem given with great accuracy. The former we identify by aid of a note on page 102, with the memoir published in the Berlin Memoir es for 1754 ; the latter is probably the memoir published in the Paris Memoires for 1758: we shall notice these memoirs in Arts. 546 and 553. The earliest entry in La Caille's journal which suggests the measurement of an arc of the meridian is dated September, 1751 : see page 144 of the book. 350 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 541. The length of a degree of the meridian assigned by La Caille was alsvays perplexing to theoretical investigators, being apparently much greater than it should have been ; at the same time, the reputation of La Caille for accuracy ensured respect for his result : see for example, Delambre's opinion, Base du Systeme Me'trique...Vol. in. page 544; and Airy's in the article on the Figure of the Earth, in the Encyclopedia Metropoli- tana, page 207. Consult also pages 463... 465 of De Zach's wo^rk, L' Attraction des montagnes.... I do not know whether De Zach ever published his promised memoir on this arc. 542. A very extensive geodetical operation has been exe- cuted in South Africa in recent times, and the results published in two quarto volumes entitled Verification and extension of La Caille's Arc of Meridian at the Cape of Good ffope, by Sir Thomas Maclear, Astronomer Royal at the Cape of Good Hope, 1866. See also the Proceedings of the Royal Society, Vol. xviii. page 109. These volumes have no Index, and no general summary of contents to guide the reader ; so that it is difficult to ensure perfect accuracy in noticing their contents. The amplitude of Sir T. Maclear's arc exceeds 4J, and the length agrees closely with the value which it should have in order to correspond with the average of the arcs measured in the Northern hemisphere : see Vol. I. page 609. The amplitude of La Caille's arc was redetermined ; the result does not differ from La Caille's by so much as half a second. Sir T. Maclear's observations were made with the zenith sector, which Bradley had used in his discovery of Aberration and Nutation ; the object glass however was not the same : see Vol. i. page 80. We read in Vol. i. page 111 with respect to the rede- termination of La Caille's amplitude : Although this work does not clear up the anomaly of LA CAILLE'S arc, yet it redounds to the credit of that justly distinguished astronomer, that with his means, and in his day, his result from 16 stars is almost identical with that from 1133 observations on 40 stars made with a powerful and celebrated instrument. MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 351 The remarkable accuracy of La Caille's amplitude would seem naturally to have suggested a recomputation of the length of his arc ; this will not be found in the volumes, and so we are left uncertain whether La Caille made some mistakes in his geodetical work, or whether the amplitude owing to deviations of the pen- dulum really was greater than corresponds to the terrestrial arc. There are indications that some investigation on this point was contemplated ; see Vol. I. pages 232, 403, 452, 456 : it is much to be regretted that this interesting question was not settled. We learn the nature of La Caille's northern station from Vol. I. pages 39, and 403 ; we are told of a mountain about half a mile distant which is not less than 2500 feet high. Reference is made to the attraction of Table Mountain, Vol. I. pages 3 and 83 ; but the subject does not seem to have been followed up afterwards. A letter however from Sir T. Maclear will be found in the Astronomische Nachrichten, Number 574, September 3rd, 1846, in which he does give some comparison between La Caille's geodetical work, and his own : the opinion is there expressed that " The chief cause of the failure of the measurement of 1752 rests with the circumstances of the terminal points." 543. In the Paris Memoires for 1752, published in 1756, there is a. memoir entitled Premier Me'moire sur la Parallaxe de la Lune...Par M. Le Francois de la Lande. The memoir occupies pages 78... 114 of the volume. The memoir discusses the observations of the Moon made simultaneously, by La Lande at Berlin, and La Caille, at the Cape of Good Hope. It touches on our subject in pages 100... 114; here we find some theory as to the evolute of the meridian, which is borrowed from Bouguer's Figure de la Terre : see Art. 363. La Lande notices three hypotheses as to the form and dimen- sions of the Earth. First, he supposes the Earth to be an oblatum in which the excentricity is . 352 MTSX'KLLANKOUS INVESTIGATIONS r.ETWKKX 174-1 AND 1760. Secondly, lie takes Bouguer's: hypothesis, tliat the increment of the length of a degree of tin- meridian varies as the fourth power of tin.- sine of the latitude. Thirdlv. he ivtuvns to the olilatuin, but applies arbitrary corrections to the three measured decrees of meridian then re- ceived: namely he adds 77 toises to the length of the decree as t O O ioiind from the arc between Paris and Amiens, and subtracts 77 toises from the Lapland degree, and he adds *2(] toises to the Peruvian degree. All that he says in justification of this process is on his page 110 : 11 me paroit d'abord naturel de supposer dans les mosuros faites ait PLTOII, line envur qui no soit quo le tiers de celle que je siipposorai dans le degre de Lapponie ot dans celui de Paris a Amiens, puisqiie dans cos deux derniers <m n'a merlin' qu'iuie amplitude d 'un degre, tandis qu'au Perou Tare se trouvcde trois degres. et inesure avec differens instrumens. By these changes, La Lande obtains for the ratio of the axes 1 2.') 2 of his ellipse _, - , -which he savs is nearly : - , and so does not H)4o " 238 229 differ much fr<jin Newton's value, namely ~. - . La Lande's fraction should be - . i'DU-to There is an account of the memoir on pages 103. ..110 of the historical portion of the volume of Memoires. We have only to notice an important error on page 10S; here it is stated that La Lande had to apply some rather large corrections to the lengths of the degrees of the meridian to make them fit Bouguer's hypo- thesis ; whereas it really was to make them fit with the figure of an oblatum. L; Lande seems to have viewed his arbitrary corrections with some satisfaction, for he refers to them about 40 years later : see his AxtroiHrtuic, \7 ( .)-, Vol. in. page :]-2. ')[{-. La Lande's second memoir on the Parallax of the Moon, is in the Pari> M>'mnircn lor 17-"->, publisheil in 17-">7. Here, La Laude continues to use LoiiguerV hypothesis; and he also MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 353 takes another modification of the elliptic hypothesis founded on the arcs in Lapland and Peru, from which he gets ^- for the ellipticity. There is an account of this memoir on pages 225... 228 of the historical portion of the volume ; reference is again made to the first memoir, without the error which occurs in the his- torical portion of the volume for 1752. 545. A memoir by Euler appears in the Berlin Memoir es for 1753, published in 1755, entitled Siemens de la Trigonomt- trie spMroldique tirts de la M&hode des plus grands et plus petits. The memoir occupies pages 258... 293 of the volume. The memoir may be said to consist of two parts. In the first part, Euler takes the lengths of a degree of the meri- dian as determined in Peru, South Africa, France, and Lapland. He assumes that there are errors in all the measures, and by means of arbitrary corrections he adjusts the lengths to coincide with the ellipticity ^ obtained by Newton from theory. Euler's cor- rections increase the Peruvian degree by 15 toises, and the French degree by 125 ; they dimmish the African degree by 43 toises, and the Lapland degree by the same amount, supposing here no allowance to be made for refraction. I presume that this is the memoir which Boscovich had in view, though the numbers are rather less extravagant than Boscovich stated : see Art. 483. In the second part of the memoir, Euler gives approximate investigations respecting the shortest line between two points on the surface of an ellipsoid of revolution. He suggests a method of using such a line for determining the Figure of the Earth: the angles which the line at its extreme points makes with the me- ridians are to be observed. But at the end of the memoir, Euler admits that the method could not practically be applied. 546. In the Berlin M&noires for 1754, published in 1756, we have a memoir by La Caille, entitled Edaircissemens sur les erreurs quon peut attribuer a la mesure du degrt en France, entre Paris et Amiens. The memoir occupies pages 337... 346 of the volume. T. M. A. 23 354 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. La Caille strenuously defends the French measurement from the charge of serious error which Euler had in fact brought against it in the Berlin Memoires for 1753: see Art. 545. La Caille is willing to stake his reputation on the statement that there cannot be an error of from 12 to 15 toises in the distance which had been determined between Paris and Amiens. A few explanatory sentences by Euler are given on the last page of the memoir. There is an allusion to the memoir in the Base du Systeme Metrique, Vol. in. page 543. 547. In the fourth volume of the Commentarii Soc. Reg. Gottingensis 1754, we have a memoir entitled Succinctam attrac- tionis historiam, cum epicrisi, recitavit Sam. Christ. Hollmannus. The memoir occupies pages 21 5... 244 of the volume. This memoir is not mathematical, and so does not fall within our range. The author holds that the word attraction is am- biguous, that Newton himself did not always use it in the same sense, and that it ought to be abandoned. He says on his last page: . . . illi, qui hac attractionis voce illudantnr, intelligere et explicare sibi posse videantur, quse neque ipsi intelligant, neque explicare aliis valeant ; . . . 548. I have alluded in Art. 301,. to the memoir by Euler on the equilibrium of fluids, which appeared in the volume of the Berlin Memoires for 1755, published in 1757. This memoir is of essential importance in the history of Hydrostatics ; but it is not necessary in connexion with our subject to give an account of it. 549. In the Paris Mtfmoires for 1755, published in 1761, we have a memoir by La Caille, entitled Sur la precision des Mesures gtodfaiques faites en 1740, pour determiner la distance de Paris a Amiens ; d V occasion dun Mdmoire de M. Euler ins6r6 dans le neuvibme tome de VAcaddmie de Berlin. The memoir occupies pages 53... 59 of the volume. This memoir resembles that which we have noticed in Art. 546, but is not identical with it. La Caille strenuously defends the MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 355 accuracy of the operations which had been mainly performed by himself. He is convinced that there is no distance on the Earth more correctly determined than that between Paris and Amiens, in which there could not be 10 toises of error. La Caille's confidence has been justified since : see Base du Systeme Metrique...Vol. in. page 162. 550. In the Philosophical Transactions, Vol. 49, Part II. which is for 1756, published in 1757, we have an Extract of a Letter of Mons. la Condamine, F.R,S. to Dr. Maty, F.R.S. trans- lated from the French. It occurs on pages 622... 624. This is a fragment of no great importance ; among other matters, it touches on our subject. La Condamine says that La Caille's measure, and that of Maire and Boscovich, do not agree with the elliptical curve of the meridian, or with the circu- larity of the parallels. He thinks that the Earth has immense cavities, and that it is of very unequal density ; consequently its figure is a little irregular. 551. We have stated that a base near that of Picard wag measured five times in 1740, and that the conclusion was drawn that there had been an error in Picard's original measure : see Art. 236. The subject was however again brought into discussion, apparently owing to an opinion expressed by Le Monnier in favour of Picard's result. The Paris Academy accordingly appointed two companies, each of four members, to test the operations. One company consisted of Bouguer, Camus, Cassini de Thury, and Pingr ; the other company consisted of Godin, Clairaut, Le Monnier, and La Caille. Each company worked independently ; and the proceedings were reported in two volumes published in 1757. I have not seen these volumes. The report of the first named com- pany is however reprinted in the Paris Memoir es for 1754, pub- lished in 1759: it occupies pages 172... 186 of the volume, and there is an account of it on pages 103... 107 of the historical por- tion of the volume. The result was a decisive confirmation of the accuracy of the operations of 1740, and consequently of the error of those origi- 232 356 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. nally made by Picard. See La Lande's BiUiographie Astronomique, page 462. We may observe that the Toise of the North and the Toise of Peru were both employed in the course of the operations ; the former appeared to be very slightly shorter than the latter. 552. In the Philosophical Transactions, Vol. 50, Part II., which is for the year 1758, and was published in 1759, there is a memoir by Charles Walmesley entitled Of the Irregularities in the motion of a Satellite arising from the spheroidical Figure of its Primary Planet. The memoir occupies pages 809... 835 of the volume. All that we have to notice in this memoir is the investigation of the attraction of an ellipsoid of revolution on a distant particle. The ellipsoid is supposed to differ but little from a sphere, and the investigation is approximate. The attraction of a sphere is known, so that we have only to find the attraction of the difference be- tween the ellipsoid and the sphere described on its axis as dia- meter. By cutting this sphere by planes at right angles to the axis, we divide it into circular rings. Accordingly Walmesley first finds the approximate value of the attraction of the perimeter of a circle on a distant particle, and then applies his result to each element of the sphere. Let a be the equatorial radius, and b the polar radius ; let f and f denote the corresponding coordinates of the distant particle : put .S 2 = 2 + f 2 . Then Walmesley obtains the following expres- sions for the component attractions of the shell parallel to the directions of and f respectively : (4 _ 4 V_ 2W J3 5tf + R* J ' 27ra(a-)6g ]4 2 i 2 ~1T~ (3 + 5 &- Walrtiesley adds a corollary on his page 815 which deserves to be noticed. I adapt his words to my own notation. He says then that the former force is to the latter as 4 4 b* 2 2 6 2 . . 4 2 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 357 He adds that if the former is represented by f, the latter must be 3& 2 represented by f ^ ; and so the resultant of the two does not O.il pass through the centre of the ellipsoid, but crosses the plane of Qfc^2 the equator at a point distant -^- towards the attracted particle. To obtain this result we must find the value of 4 2 3 + 5 2^ 2 | ' i? J . 4 4 fc1 f 3 ft' 3fW- 1 this is fjn-ioap-'ttf i 1 -*^* a?) ' that is approximately 35 s that is 553. In the volume of the Paris Mtfmoires for 1758, pub- lished in 1763, there is a memoir by La Caille, entitled Me'moire sur la vraie longueur des Degrtfs du Meridien en France. The memoir occupies pages 237... 244 of the volume. The astronomical observations in the work entitled La Meri- dienne de Paris verifie'e, 1744, had not been corrected for what we now call Nutation. This irregularity had been discovered before that work appeared, but the theory had not been published ; and it was supposed that the error produced during the interval of sixteen months, over which the operations extended, might be neglected. La Caille now applies the proper corrections to the amplitudes, and to the deduced lengths of degrees of the me- ridian. 554. The volume of the Paris Me'moires for 1759J published in 1763, contains a memoir entitled Mtmoire sur les Degrfa de 358 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. I' ellipticite' des Sphtfroides, par rapport a I'intensite' de T attraction. ParM.le Chevalier D'Arcy. The memoir occupies pages 318... 320 of the volume. We may say, in modern language, that this short memoir draws attention to the principle of the conservation of areas, as holding in the case of a mass set in rotation, and acted on by no forces except the mutual attractions of its particles. The writer calls the principle the conservation of action, and claims it for his own. See Walton's Mechanical Problems, 1855, page 479. Laplace gives an application of the principle in the Mecanique Celeste, Livre in. 21 : see Art. 286. 555. The Academy of Toulouse proposed the Figure of the Earth as the subject of an Essay, with a double prize, for the year 1750. The prize was obtained by Clairaut. The volume con- taining the Essay appears to have been published at Toulouse in 1759. I have not seen this volume. There is an account of the Essay in the Journal des Spavans for October 1759; this account occupies pages 281. ..301 of the Amsterdam edition of the volume of the Journal. The account is obscure and uninteresting, like most of the attempts to translate mathematical investigations into ordinary language. Hence I do not submit with much confidence the following brief notice of what Clairaut' s Essay seems to have contained. Clairaut considered that both the ellipticity of the Earth, and Clairaut's fraction, were found by observation to be greater than they would have been for a homogeneous fluid. Hence Clairaut's theorem does not hold for the Earth ; and so it becomes neces- sary to devise some hypothesis which differs from those on which that theorem may be established. Clairaut first examines an hypothesis which he attributes to Bouguer ; namely, that the parts of the Earth in the vicinity of the axis of rotation are denser than the rest of the Earth. Clairaut comes to the conclusion that this is inadmissible. He finds that if the density in the vicinity of the axis differs from the density of the rest of the earth, it will not be possible to obtain an ellipticity and a Clairaut's fraction which shall both be greater than for a MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 359 homogeneous fluid. We are not referred to the place where Bouguer has maintained this hypothesis. Then Clairaut proposes his own new hypothesis. He assumes a solid nucleus. The generating curve is to differ slightly from an ellipse ; every ordinate exceeding the corresponding ordinate of the ellipse by a small quantity which varies as the cube of the cosine of the latitude. Thus, in addition to the attraction of an ellipsoid of revolution, he has to consider the attraction of a certain shell which is also a figure of revolution. By investigating the problem, and following the hints which may be extracted from the Journal des Sgavans, it will be seen that Clairaut's processes, though tedious, would not have involved any very serious difficulty. 556. La Lande mentions this Essay: see his BibliograpMe Astronomique, page 464. He ascribes the account in the Journal des Sgavans to Clairaut himself, "... ou Clairaut en donna lui- meme 1'extrait." But from the commendation bestowed on the Essay in the account of it, I think that La Lande must be wrong. It is dif- ficult, for example, to believe that Clairaut could have praised himself in these words : Toutes ces transformations que nous indiquons, et que M. Clairaut emploie avec taiit d'art et de succes, doivent etre regardees comme le sceau du Geometre superieur qu'il imprime a tous ses Ouvrages. The Essay can be regarded only as a mathematical exercise ; and it does not seem ever to have attracted attention. It is not mentioned in the translation of Newton's Principia, which was prepared by Madame du Chastellet under the guidance of Clairaut; nor in Poisson's reprint of Clairaut's Figure de la Terre : in this reprint some account of the Essay might with advantage have been given. 557. A problem in the Integral Calculus is mentioned with approbation ,as the foundation of many of Clairaut's investigations : see page 296 of the account in the Journal des Sgavans. I will 360 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. endeavour to reconstruct this problem from the obscure traces which are given. The equation to the generating curve of the nucleus which Clairaut adopts will be of the form (1). Let this curve revolve round the axis of x, so that the equa- tion to the nucleus is <) = -V(a 2 -* 2 )+Ma 2 -* 2 ) .......... (2). CL "We suppose \ so small that its square may be neglected. The area of a section of the solid made by a plane at right angles to the axis of x, and at the distance x from the origin, will be try*, that is by (1) approximately if -t (a 8 - x*} + 27r\ - (a 2 - xj. a a It is required to shew that the area of a section made by a plane at right angles to the axis of y, and at the distance y from the origin, can be put in an analogous form. We have from (2) ' 4 x > /Xl/ A x Thus The area of the section to our order of approximation will be re ^ 4 I zdx, where c stands for -n MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 3G1 Hence by assuming & = -& V(-^ 2 ~ #*) sm $ we easily find that the area is This expression is of the required form. The use of the problem to Clairaut consists in this : as the formulas for the attraction of an ellipsoid of revolution may be con- sidered known, as soon as he has determined the attraction of the nucleus for a point on the axis of x, he can readily infer the attraction for a point on the axis of y. 558. A celebrated French lady translated Newton's Principia and added a commentary; the work was published after her death under the title of Principes Mathtfmatiques de la Philoso- phic Naturelle, par feue Madame la Marquise du Chastellet. 2 vols. 4to. Paris, 1759. Besides Chastellet we have the variations Chastelet and Chatelet: see pages iv and v of the first volume of the work. The work has an Avertissement de I'Editeur, and a Preface Histo- rique by Voltaire. From these it appears that Madame du Chastellet was a pupil of Clairaut's; and the commentary was constructed out of the materials which she obtained from him. The translation occu- pies the first volume and part of the second; the commentary occupies the remainder of the second volume. We will notice those pages of the commentary which bear on our subject. 559. Pages 56... 67 give an analysis of Newton's method of treating the Figure of the Earth. On page 62 the cause of a mistake made by Newton is assigned as in Clairaut's Figure de la Terre, page 256 ; though here apparently with more confidence : see Art. 37. On page 66 the criticism on Newton's conjecture with respect to Jupiter is given as in Clairaut's Figure de la Terre, page 224 ; though here apparently with more confidence : see Art. 31. The pages 155... 183 constitute an analytical treatise on At- 362 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 17GO. tractions in three sections. First we have spherical bodies, then bodies of other forms, and lastly an ellipsoid of revolution with the attracted particle on the prolongation of the axis. The in- vestigations are simple and satisfactory. One of these investigations relates to the attraction of a sphere on an internal particle, when the force varies inversely as the fourth power of the distance. To avoid the infinite expressions which might occur, it is assumed that if the particle be at the centre of such a sphere, the resultant attraction must be zero. The pages 193... 259 form a section entitled De la Figure de la Terre, in two parts. The first part on pages 193... 221 is an abridgement of the theory of Hydrostatics which constitutes the first half of Clairaut's volume. The second part, on pages 221 ... 259, is on the Figure of the Earth ; this is almost a reproduction of Chapters II. and ill. of the second half of Clairaut's volume. Two simple examples constitute all* the novelty which the commentary furnishes ; we will mention these. On page 238 the general formula for the value of attraction given by Clairaut on his page 247 is applied to the case in which the strata are all similar, so that the ellipticity is constant, and the density varies as the distance from the centre : see Art. 336. The result is found then to be independent of s, so that the attraction is approximately constant at all points of the surface : see Case in. of Art. 266. The result in our notation will be found to be 2 Trrf\ (l + o 6 i)> where X is the density at the unit of distance o from the centre. The other example is discussed on pages 238, 239 and 246. Using the notation of Art. 336, suppose that the density and the ellipticity are given by these formulae p=\r^pr t *<>*, ' i where X and p are constants. Then the commentary finds the expression for gravity and the value of e x . See Arts. 336 and 327. 560. According to the Preface Historique, page ix, great care was bestowed on the Commentary. When the lady had written a MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. 363 chapter, Clairaut examined and corrected it ; and subsequently the fair copy Avas revised by a third party. Nevertheless there are numerous errors or misprints, some of which are very serious. Thus, for example, the formulas which occur in the investigation of the attraction of an oblongum on pages 182 and 183 are much disfigured ; the former part of page 197 is unintelligible, owing to the omission of important matter ; the binomial expressions on pages 218 and 219 are extremely inaccurate ; the reference to a supposed property of the ellipse on page 242 is absurd ; and the numerical application of Clairaut's Theorem on page 257 is quite wrong. Playfair mentions Madame du Chastellet in his Dissertation... of the Progress of Mathematical and Physical Science... ; see page 655 of the Encyclopedia Britannica, eighth edition, Vol. I. In reference to her writings on the dispute as to the measure of force, he says : ...from the fluctuation of her opinions, it seems as if she had not yet entirely exchanged the caprice of fashion for the austerity of science Voltaire however finds merit in a similar fluctuation. He says in the Preface Historique, page VI. : ...Ainsi, apres avoir eu le courage d'embellir Leibnitz, elle eut celui de 1'abandonuer : . . . Playfair speaks highly of the translation of Newton and the Commentary. I do not agree with his estimate of the Com- mentary. The title is really inappropriate. Instead of any ex- planation of Newton, we have merely other investigations drawn from well known works exhibiting more recent solutions of the problems which Newton discussed. 561. The first volume of the series published by the Turin Academy which is usually called the Miscellanea Taurinensia is dated 1759. On pages 142... 145 Lagrange supplies a note to a memoir by another person. The note relates to the attraction of an indefinitely thin spherical shell, and undertakes to explain the paradox which had disturbed D'Alembert : see Art. 434. Lagrange's explanation is rather a hint than a strict mathe- 364 MISCELLANEOUS INVESTIGATIONS BETWEEN 1741 AND 1760. matical investigation ; but the idea is sound and valuable. When the attracted particle is very near the shell, an infinitesimal part of the shell close to the particle produces a finite portion of the whole attraction, in fact a half. When the attracted particle forms an element of the shell, this part of the attraction vanishes ; and when the attracted particle is inside the shell it becomes negative. Lagrange's idea for the spherical shell is really the same as Coulomb afterwards used for a shell of any form in the Paris Mtfmoires for 1788 ; Laplace developed it in an investigation which occurs in Poisson's memoir on the distribution of electricity, in the Paris Memoir es for 1811. 562. The second volume of the Miscellanea Taurinensia, is for 1760 and 1761 ; the date of publication is not recorded. This volume contains the memoir by Lagrange on Maxima and Minima, which is famous in the early history of the Calculus of Variations, In a memoir immediately following, and connected with this, Lagrange treats of various problems in Dynamics ; and* among others, he considers the motion of fluids. Lagrange, on his page 282, makes a remark respecting a passage in D'Alembert's...jR&istoce des Fluides to which I have already alluded: see Art. 397. Lagrange makes a remark that surfaces of equal density will be level surfaces provided a certain condition holds. He says on his page 284 : Cependant un grand Geometre a cru que il n'etoit pas toujours ne"cessaire que les surfaces des differentes couches fussent de niveau, et il a donn6 un autre Principe pour connoitre la figure de ces surfaces. Lagrange commences a mathematical investigation ; and in effect he says that if we proceed according to D'Alembert's man- ner, as given in Art. 401, we shall find that the surfaces of equal density are level surfaces. D'Alembert as we have stated, subse- quently admitted his error: see Arts. 368 and 400. Lagrange criticises other opinions of D'Alembert on pages 275 and 323 of the second volume of the Miscellanea Taurinensia : but these do not belong to our subject. CHAPTER XVI. D'ALEMBERT. 563. WE shall now resume our examination of the labours of D'Alembert in our subject. With a few unimportant exceptions, the present Chapter will be devoted to memoirs published by D'Alembert in various volumes of his Opuscules Mathe'matiques. 564. The article entitled Figure de la Terre in the Encydo- pe f die, was by D'Alembert ; the date of the volume in which it was published is 1756. The article occupies pages 749. ..761 of the volume; it gives an interesting account of the measurements and of the theoretical investigations on the subject up to the date of publication. D'Alembert awards high praise to Maclaurin, and to Clairaut ; and refers with obvious satisfaction to his own researches. He notices especially the Articles 16 6... 169 of his Essai sur la R6- sistance des Fluides ; these he appreciates at a value far beyond their worth : see Arts. 404... 406. He also refers to his Recherches . . . Systeme du Monde, and it may be admitted that these volumes are not without merit as regards our subject. D'Alembert discusses at some length in a popular manner the question as to whether the Earth can be assumed to be a figure of revolution. In the Encyclope'die Mtihodique the account which D'Alembert gave of the theory of the subject is reproduced ; but his account of the measurements is omitted, and a shorter article respecting them by La Lande is supplied. In the Encyclope'die Me'thodique there is a reference to the fifth volume of the Opuscules Mathe'matiques which of course was not in the original Encyclope'die. 366 D'ALEMBERT. The following sentence of the original article is worth notice : ...ceux qui les premiers mesurerent les degres dans 1'etendue de la France, preoccupes peut-etre de cette idee, que la Terre applatie donnoit les degres vers le nord plus petits que ceux du midi, trouverent en effet que dans toute 1'etendue de la France en latitude, les degres alloient en dimmuant vers le nord. In the original article speaking of what his HechercTies . . . con- tained D'Alembert says : il pourroit tres-bien etre en e'quilibre sans avoir la figure elliptique. This is not so strong as the preface to Vol. ill. of the Itecherches...p&ge xxxvi. 565. The article entitled Gravitation in the Encyclopedic was by D'Alembert ; the date of the volume in which it was published is 1757. The only part of the article which concerns us consists of some observations respecting the paradox which D'Alembert considered that he had discovered as to the attraction of an infinitesimally thin spherical shell: see his Recherches...Systeme du Monde, Vol. III. page 199 ; and Arts. 434 and 561. D'Alembert shews analytically, that if a particle be outside the shell the resultant attraction on it is the same as if the mass of the shell were collected at the centre ; this result is no more than Newton had given in his Principia: see Art. 4. * D'Alembert's observations to which we are here referring are omitted in the article Gravitation of the Encyclopedic Methodique; but the substance of them is reproduced as we shall see in the first volume of the Opuscules Mathe'matiques. 566. The first volume of D'Alembert's Opuscules MatMma- tiques was published in 1761. On pages 246... 264 we have a memoir entitled Eemarques sur quelques questions concernant T attraction. 567. According to D'Alembert's formula on page 42 of his Reflexions .. .des Vents, which is reproduced in our Art. 376, rela- tive equilibrium might subsist in the case of a solid oblatum covered with fluid, such that the external surface of the fluid was an D'ALEMBERT. 3G7 oblongum; the whole rotating on a common axis. This result, D'Alembert says, had been attacked by un Gtometre Italien, qui a du nom dans les Mathematiques : pages 246... 25 2 constitute a reply to this attack. The Italian Geometer was doubtless Boscovich ; see page 463 of his De Litteraria, Expeditione..., and Art. 470. The objec- tion urged against D'Alembert's result amounts to this in modern language: that the relative equilibrium would not be stable. D'Alembert says, very justly, he might reply that in such researches no mathematician had as yet attempted to consider whether this condition was satisfied. However, he makes some remarks on the point. He contents himself with shewing when the tangential force at the external surface of the fluid would act towards the pole, and when towards the equator. He arrives at the conclusion that the relative equilibrium would be stable in the case to which objection had been taken, provided the density of the fluid were less than five-thirds of the density of the solid. D'Alembert's investigation is not adequate to solve the problem of the motion of the fluid when disturbed from its position of relative equilibrium : but his defence is at least as good as the attack of the Italian Geometer. The subject will appear before us again in the sixth volume of the Opuscules Mathematiques. 568. On his pages 252... 257, D'Alembert corrects an error into which he had fallen in his Reflexions .. .des Vents, pages 15 5... 15 7. We have already paid attention to this correction: see our Art. 381. 569. On his pages 257, ..264, D'Alembert recurs to what he considered a paradox, as to the attraction of an infinitesimally thiu spherical shell : see Art. 565. D'Alembert reproduces the sub- stance of some remarks originally published in the Ency dope' die. He takes objection to Lagrange's explanation, and he says he gives one of his own : see Art. 561. What D'Alembert really does is to translate Lagrange's idea from popular language to mathematical language ; and then to ascribe the entire merit to himself. He shews that the infinitesimal part of the shell to which Lagrange 368 D'ALEMBERT. refers may be taken to be the part determined by tangents from the external particle. It may be observed that neither Lagrange nor D'Alembert uses a symbol to express the infinitesimal thickness of the shell. If we consider the shell to be very thin, though not infmitesimally thin, and suppose the attracted particle to pass gradually from the outside of the shell to the inside of the shell, all the so-called paradox disappears : for the attraction changes gradually and not discontinuously. 670. The fifth volume of D'Alembert's Opuscules Mathe'ma- tiques was published in 1768. On pages 1...40 we have a memoir entitled Sur Vtquilibre des Fluides ; and the pages 23... 40 of this memoir constitute an Appendice sur la Figure de la Terre. 571. The memoir begins by corrections of errors in preceding investigations. D'Alembert had supposed that he had obtained in his Essai sur la Resistance des Fluides more general results than his predecessors in the theory of the equilibrium of fluids and the figure of the Earth. He now admits that the supposition was unfounded. The quantity denoted by K on page 210 of that work he now allows should be zero; and so his result coincides with Clairaut's: see Art. 405. In like manner he admits that the same simplification ought to be made in various equations which he had given in the third part of his EechercTies Systeme du Monde, beginning with page 229. I have already, in my account of this work, noticed the correction : see Art. 444. 572. D'Alembert returns to the subject he had introduced on page 203 of his Essai sur la Resistance des Fluides : see Art. 400. He maintains, and rightly, that in a fluid in equilibrium the sur- faces of equal density are not necessarily level surfaces. He ad- mits, however, that for such forces as occur in nature, the surfaces of equal density are level surfaces; his original error on this point was corrected by Lagrange : see page 2 of the Opuscules Mathtmatiques, Vol. V. ; also Arts. 405 and 562. D'ALEMBERT. 369 573. D'Alembert gives a form, at once simpler and more general, to the equations which he had used in the Essai sur la Resistance des Fiuides : see page 6 of the Opuscules Maihgma,- tiques, Vol. v. ; also Art. 402. 574. D'Alembert occupies his pages 10... 22 with remarks on the conditions of fluid equilibrium. The remarks are sound, must have been valuable at the time, and may even now be read with profit. D'Alembert objects with justice to Clairaut's expla- nation of a paradox in the subject ; see Art. 312. The main principle which D'Alembert asserts, expressed in modern language, is in effect this : Consider only forces in one plane; then we have for the equilibrium of a fluid the equa- tions Take the simple case of homogeneous fluid; then it is not sufficient for equilibrium that Xdx 4- Ydy should be a perfect differential. If we suppose that Xdx + Ydy is the differential of <f) (x, y) then < (x, y) must have only one value for given values of x and y. Thus, for example, </> (x, y) must not be such a function as tan' 1 - . x Again, suppose we use polar coordinates, and find that p = F(r, 0) ; then when r = we have p apparently a function of 6 only. But unless this apparent function of 9 reduces to a constant, the pressure would not be the same in all directions about the origin; which is contrary to the nature of a fluid. In the two preceding paragraphs we have translated D'Alembert's ideas into modern language ; he himself does not speak of pressure, nor does he use the symbol^?. 575. D'Alembert devotes his pages 23... 40 to an Appendix on the Figure of the Earth. His object is to enquire if the obla- tum is the only form of relative equilibrium for a rotating mass of homogeneous fluid. He says it follows from what he has proved in his Eecherches sur les Vents, Art. 28, that if the fluid mass is originally spherical, T. M. A. 24 370 D'ALEMBERT. and is then put into rotation, so that the ratio of the centrifugal force to gravity is small, the form of relative equilibrium must be an oblatum. It is almost needless to remark that D'Alembert's statement is not demonstrated : the motion of such a fluid mass is too difficult for his rough approximative analysis to master. However, he now proceeds to discuss the problem without assuming that the mass is originally spherical. He arrives at the conclusion that if the fluid is in the form of a figure of revo- lution, and is nearly spherical, there cannot be relative equilibrium for any other form than an oblatum. Unfortunately, his demon- stration is unsound. The theorem, however, is now admitted to be true. Legendre indeed gave a demonstration, which does not assume the figure to be nearly spherical ; but the demonstration is not quite free from objection. Laplace, assuming the figure to be nearly spherical, but not assuming it to be of revolution, demonstrated the theorem : he omits to mention the condition that the figure is nearly spherical when he refers to the subject in the MJcanique Celeste, Yol. v., page 10. 576. We will now indicate the nature of D'Alembert's method, and the point at which it fails. Let P be the pole of the body, Q any point on the surface. We shall require the attraction at Q resolved in the direction which is in the meridian plane of Q, and is at right angles to the radius from the centre to Q. Let R be any point on the surface ; let PQ = ft QR - u, z\ and let PQR = 7r iJr, so that ty is the angle between RQ and PQ produced. Let the polar radius be denoted by 1, D'ALEMBERT. 371 and the radius at E by 1 +aF(z), where a is a small quantity. Then proceeding as in Art. 424, we find that the element of the required attraction, estimated from the pole, to the order we have to regard du dty sin u cos ^ cos J u ^ . . 4sin a iw _ du d-^r sin 2 u cos ^ 8 sin 3 u * ' _ du dty sin 2 u cos -/r 2 f (l-cos<w) f This agrees with D'Alembert's formula at the top of his page 26 ; his A is our ^. The transverse attraction which we require would be obtained by integrating the above expression between the limits and 2-7T for i/r, and and TT for u. Let T denote this transverse attrac- tion. Let V denote the attraction at Q resolved along the radius ; and % the angle between this radius and the tangent to QP at Q. Then V cos ^ is the resolved part of F along the tangent to QP at Q. Hence, supposing the body to be fluid, or at least the outer stratum to be fluid, we must have for equilibrium Fcos % =T .............................. (1). If the body rotates, then to secure relative equilibrium, we must supply in this equation a term corresponding to the resolved centrifugal force. We must now give some specific form to F(z) before we can carry the investigation further. Assume, with D'Alembert, that F(z) =A + B cos^+ Ccos*z + ...... + M cos m z. We shall then have to our order of approximation = - sin# (# -1-2(7 cos /3+ ..... , + mM cos" 1 ' 1 /3) a ; and it will be sufficient in (1) to put r for F. Thus (1) becomes o mM co^ 1 0) = T... ..(2). 242 372 D'ALEMBERT. Now cos z cos (3 cos u sin ft sin u cos ty. Hence, correspond- ing to the term Mcos m z in F(z) we have in jTthe term f *" f 2ir sin 2 u cos -\lr (cos j3 cos u sin /3 sin u cos \Jr) t!l , .. , - I - - - -TZ - ^ - r ^- du d^r. Jo Jo (1 -COB ii)l When we integrate with respect to ty all the terms which in- volve odd powers of cos ty vanish ; so that we are left with aHf C v f 2 "" sin 2 ^- "sTJoJo HI where Z m cos" 1 " 1 ft sin 8 cos Here every term involves some odd power of sin /3. Now suppose we put (1 -cos 2 /3) sin/3 for sin 3 /3, and (1 -cos 2 /3) 2 sin /3 for sin 5 /?, and so on. Then Z takes the form sin 3 cos 7 "' 1 where -A^, ^7" 8 , JV B , ... are functions of u and ty. In like manner the other terms in F(z) will give rise to cor- responding terms in T, involving the product of sin ft into various powers of cos ft ; but these powers of cos/3 will all be less than the (m l) th power. Hence equating the coefficients of like terms in (2), we see that besides other relations we must have _ 4-Trm _ 1 /"* T 2flr JVj sin 2 u du dty ~3~ ~2^ JoJo (1 - cos M)* D'Alembert then has to shew that equation (3) cannot be satisfied if m be a positive integer greater than 2. His demon- stration, however, fails completely, because he has given a wrong value to the quantity which we denote by JV t : his error begins with his Article 50, on his page 26. D'ALEMBERT. 873 Take the second term which we have expressed in the value of Z\ and put sin/3(l cos 2 /3) for sin 3 /?: then we have as part of NI If Instead of keeping this, D'Alembert puts sin u (1 cos 2 u) for sin 8 M, and then omits sinw, retaining only sinw cos 2 w, so that instead of what we have just given, he has m (m 1) (m 2) _, , - gJ - ' cos" 1 u sm u cos ^. He treats the other terms of Z in the same unwarrantable manner; and the consequence is that his value of N^ is alto- gether wrong. The error renders all the rest of his argument worthless. Laplace, as we shall see, alludes in his first and second memoirs to D'Alembert's demonstration, but says nothing about its un- soundness. Legendre, who may be considered to have been the first to solve the problem here involved, does not even allude to D'Alembert's demonstration. 577. It is important to notice what D'Alembert's process would have established if it had been sound. It would have shewn that F(z) cannot be a finite series of powers of cos z, in which the highest power is greater than 2. But it would not have shewn that F(z] cannot be an infinite series of powers of cos z. 578. D'Alembert gives on his page 29 the value of the de- r-l x m-l ^ x finite integral I ^ when for m we put 1, 2, 3, 4, 5, 6, or 7. J* (1 - 'mf There is no objection to his method. We may, if we please, rv 2 transform the integral to 2 (- l) m (y z -l} m ~ l dy : thus it is easy J o to verify his values. After leaving this subject, D'Alembert on his pages 36... 40, makes a few other remarks ; they are not of great importance, but 374 D'ALEMBERT. they are correct, except those contained in his Art. 78, which are erroneous. 579. The sixth volume of D'Alembert's Opuscules matiques was published in 1773 ; a large part of it is devoted to our subject. 580. A memoir entitled Sur la Figure de la Terre, occupies pages 47... 67. It begins thus : Feu M. Maclaurin est le premier qui ait d6montr rigoureusement qu'une masse fluide homogene, tournant autour d'elle-mcme, devoit prendre la figure d'une ellipse dans 1'hypothese de 1'attraction en raison inverse du quarre des distances. Mais personne, qne je sache, n'avoit encore remarque que dans ce cas le probleme est susceptible de deux solutions, c'est-a-dire, qu'il y a deux figures possibles a donner au sphe- ro'ide, et dans lesquelles 1'equilibre aura lieu. Cette consideration est 1'objet des Recherckes suivautes. We have already remarked that Thomas Simpson had im- plicitly shewn the possibility of this double solution : see Art. 285. However, D'Alemtert now gives an explicit investigation, which, in substance, was afterwards incorporated by Laplace in the Mecanique Celeste, and thus constitutes a permanent part of the subject : see the Mecanique Celeste, Livre in., Chapitre ill. On his page 47, D'Alembert makes the undemonstrated asser- tion, that if a spherical mass of homogeneous fluid be put in rotation it will take the form of an oblatum : see Art. 575. 581. We will, in giving an account of D'Alembert's process, adopt to a great extent Laplace's notation. Suppose a) the angular velocity of rotation, p the density of the fluid ; put q for - . Suppose the major axis of the Earth to be 1) times the minor axis; then the excentricity of the ellipse is . . : . Denote the minor axis by 2c. Then by the VIA ' -V D'ALEMBERT. 375 formulae of Art. 261, or by those of any elementary work on Statics we find that the attraction on a particle at the pole is 47rpc(l+X 2 ) f tan" 1 *,) "XT' T- "1TT and the attraction on a particle at the equator is 27rpcV(X 2 +l) f tan^X 1 ^- -|(i- *0-x~ 'T call the latter X, and the former Y. The centrifugal force at the equator Then, as in Art. 262, the condition for relative equilibrium is which reduces to 2q _ (X 2 + 3) tan" 1 X - 3X ( . 3 = X 3 This is the standard equation on the subject: see the Me'ca- nique Celeste, Livre ill. 18. D'Alembert has the same equation: see his page 50. He uses k for Laplace's X, and o> for Laplace's q. Neither D'Alembert nor Laplace uses the symbol tan" 1 , which is of more recent origin. It will be observed that if a sphere of the same density as the fluid were to rotate with the same angular velocity, q would be the ratio of the centrifugal force to the attraction at the equator. . D'Alembert shews that the equation (1) will give two values of X for a given value of q, provided q be not too great. Denote by $(X) the right-hand member of equation (1) ; then consider- ing X as an abscissa, and <(X) as the corresponding ordinate, he in fact traces the curve which thus arises. We have <(X) zero when X is zero, and also when X is infinite ; when X is very small 4X 2 is approximately equal to ^-r-. Since $(X) vanishes when 376 D'ALEMBERT. X vanishes, and when X is infinite, there must be some maximum value of X ; this maximum is determined by putting <' (X) = ; this leads to 9X + 7X 3 It is evident from what has been said that this equation must have a root. We may also establish the existence of a root in the following way. When X is very small the left-hand member is approximately \8 -v6 A- 4- "3 + 5~ and the right-hand member is approximately x 7x ~3~ + 27~ thus, when X is very small, the right-hand member is the larger. When X is infinite the left-hand member is the larger. Hence for some intermediate value the two members will be equal. See D'Alembert's pages 51 and 52. 582. Suppose that q has a given value ; let \ denote the smaller of the two values which equation (1) furnishes. By corn-- paring the weights of a polar and an equatorial column of fluid, without assuming that there is equilibrium, D'Alembert finds that if X is a little less than X, the weight of the polar column predomi- nates, and that if X is a little greater than \ the weight of the equatorial column preponderates. Then he argues thus : Let the fluid be in relative equilibrium with the value X r Suppose the oblatum a little elongated ; this amounts to diminishing X ; then the weight of the polar column preponderates, and pushes out the equatorial column : thus there is a tendency to restore the equi- librium figure. Again, suppose that we start from the equilibrium figure, and compress it a little; this amounts to increasing X; then the weight of the equatorial column preponderates, and pushes out the polar column: thus there is a tendency to restore the equilibrium figure. Hence in modern language the relative equi- librium is stable ; D'Alembert uses the word ferme. D'ALEMBERT. 377 In like manner he concludes that the relative equilibrium cor- responding to the larger of the two values which equation (1) furnishes is unstable. His discussion on these points will be found on his pages 55. . .57 : it cannot be considered adequate for such a difficult matter. I do not find that the later writers Laplace, Poisson, and Ponte'coulant have followed D'Alembert in determining the stability or insta- bility. If the angular velocity is such as corresponds to a single solu- tion, so that (1) and (2) are simultaneously satisfied, D'Alembert arrives at what he considers a singular result. This result ex- pressed in modern language is that the relative equilibrium is stable with respect to an elongation of the oblatum, and unstable with respect to a compression of the oblatum : see his page 57. 583. On his page 58, D'Alembert says : Ceci me porteroit a croire, pour le dire en passant, que dans les Theories donnees jusqu'ici sur la Figure de la Terre, on a peut-etre trop cherche a faire accorder entr'eux les deux principes, celui de la perpendi- cularite de la pesanteur a la surface, et celui de I'Squilibre des colomnes. Car ce dernier n'est necessaire que quand la Terre est fluide, et n'est jamais suffisant, soit que la Terre soit solide ou fluide ; au lieu que le premier est necessaire dans les deux cas, et suffit si la Terre est solide. By the principle of columns he probably means the balancing of columns at the centre. Boscovich had shewn that if at every point every pair of rectilinear columns balances, then also Huygens's principle of equilibrium is satisfied : see Boscovich's De Litteraria Expeditione...peige 424 ; and Art. 463. 584. When the angular velocity is very small, one of the forms of relative equilibrium determined by equation (1) is very nearly spherical, and the other is very much compressed; D'Alembert calls this a singulier paradoxe : see his page 58. Let us suppose that q is very small ; one value of \ is very large as we have said. Thus (1) becomes approximately 378 D'ALEMBERT. therefore X = ................................ (3). Let r be the radius of a sphere having the same volume as the oblatum ; then with the notation of Art. 581, D'Alembert shews that the velocity of a point at the equator is very small when X is very great ; that is, the smallness of the angular velocity more than counterbalances the largeness of the radius. For the square of this velocity by (4) = i*\-q approximately this is small since q is small. D'Alembert also compares the centrifugal force at the equator in this case with the centrifugal force at the equator of the sphere of equal volume. The ratio of the former to the latter this is large since X is large. See his page 59 ; there are mis- prints towards the bottom of the page. 585. D'Alembert was aware that his investigations did not shew that there could not be more than two forms of relative equilibrium corresponding to a given angular velocity. He ex- pressly leaves this point to be discussed by other Geometers : see his page 61. Laplace was the first who demonstrated that there could not be more than two forms of relative equilibrium: see D'Alembert's Opuscules Mathe'matiques, Yol. VIII. page 292, and Laplace's Theorie...de la Figure des Planetes, page 124. 586. The proposition which D'Alembert thus left to be de- monstrated amounts to this, that </>'(X) vanishes only once as X D'ALEMBERT. 379 changes from zero to infinity, besides when X = ; D'Alembert draws his curve consistently with this proposition, though he did not demonstrate it. The proposition is known to be true as it is indirectly involved in Laplace's investigations; but it may be useful to give a direct demonstration. Put tan for X ; then (3 + tan 2 0) 0-3 tan (1 + 2 cos 2 0) 6 - 3 sin cos ~- -- The differential coefficient of this with respect to is cos0(8+cos20) (5 + 4 cos 20) 0_ sin 20 (8+ cos 20) -20 (5 +4 cos 20) ~~suf0~ sin 4 0" 2sin 4 Put F(6) for the numerator. I //T5 When is very small, we shall find that F(0) = -^-=- . This is lo easily obtained by expansion, for F(ff) = 8 sin 20 + \ sin 40 - 20 (5 + 4 cos 20) . L Or we may proceed thus : we know that when X is very small <(X) = - ? so that <'(X) then = ^; hence when is very small J.o JLo F(0) 80 .-, 160 5 we must have g-jj^ji- jj > and therefore F(Q = .-^-. When = ^ we see that -F(0) is negative. If then F(6) vanishes for more than one value of 0, besides = 0, between = and = -^ , it must vanish for three values : Z and then F'(ff) must varfish for two values of besides = 0. But F'(0) = 8 cos 20 + 2 cos 40 - 10 + 1C0 sin 20 ; F'(0) = - 8 sin 40 + 320 cos 20 = 16 cos 20 (20 - sin 20). 380 D'ALEMBERT. Thus F"(6) is positive from = to = T > an d then negative from 0=7 to = s; therefore -F'(^) increases continually from 4 .4 6 = to = -T i and diminishes continually from = ^" to 6 = : 4 42 hence ^'(^) cannot vanish more than once besides 0=0, as 8 changes from to ~ . ft 587. We may put equation (I) in the form = -. o If we suppose X = 0, both sides vanish whatever may be the value of q. But X = is not a solution of (1); we have in fact introduced this solution by multiplying both sides of (1) by X s . D'Alembert devotes his page 62 to this matter ; which would now be considered too obvious to need remark. 588. D'Alembert gives some extension to his investigation on his pages 63... 67 by supposing extraneous forces to act; but this extension is of little importance. D'Alembert afterwards returns to the subject and discusses it in an elaborate manner: see Art. 596. At the top of his page 64, D'Alembert seems to say he has four forces ; but his first force is in fact resolved into his second and third, and is not in addition to them. 589. The next memoir in the sixth volume of D'Alembert's Opuscules Mathtfmatiques is entitled Eclaircissemens sur deux endroits de mes Ouvrages, qui ont rapport a la Figure de la Terre ; this occupies pages 68... 76: it is followed by some Eemarques sur Article pre'ce'dent on pages 77... 84. The passages in his previous works to which D'Alembert here alludes occur on page 42 of the Reflexions... des Vents, and on pages 246... 252 of the first volume of the Opuscules MatM- matiques: see Arts. 376, 378, 514, and. 567. D'ALEMBERT. 381 590. We have already learned from Art. 567, that Boscovich criticised D'Alembert, and that D'Alembert defended himself. Boscovich' s work was translated into French, and a long note inserted on pages 449... 453 which renewed the attack on D'Alembert : and now D'Alembert replies. The matters in controversy admit of being stated briefly though neither of the disputants defines them very clearly. The translator ascribes great merit to Boscovich for intro- ducing the notion of what we should call the stability of the equilibrium : D'Alembert replies that the notion is really due to Daniel Bernoulli. Next as to mathematical results we may say that both disputants accepted the formula of Art. 376; and also both allowed that the equilibrium would be stable if p were less than - cr. Then D'Alembert asserts that we may have p less o than - cr, and e' positive, and yet have e negative ; and the for- mula of Art. 376 shews that his statement is correct. The French translator denies this, and so is wrong; he seems to have assumed that 1 - must be positive, which is not necessary. The following passage of the translator's note relates to the opinion which D'Alembert held of Boscovich. ...M. d'Alembert se contente ici de dire qvtil a du nom dans les mathematiques : dans un autre opuscule posterieur, il parle du P. Boscovich avec eloge, en disant qu'il merite la reputation dont il jouit ; mais pour aj outer qu'il a ete telleraent persecute par les Superieurs de son Ordre, que toute Fautorite du Souverain Pontife a a peine suffi pour le delivrer de leurs poursuites. Cependant on sait tres bien que le R. P. Boscovich a toujours etc consider^ et respecte dans sa Compagnie comme uii de ses plus dignes membres, et comme un homme du premier merite a tous 6gards. On page 71 of the memoir by D'Alembert which we are now considering he uses the words habile Math&naticien, I presume with reference to Boscovich. It has been asserted in recent times that D'Alembert and Lagrange had but a low opinion of Boscovich ; see Arago's (Euvres completes, Vol. II. page 140. 382 591. D'Alembert states on his page 75 his objection to the formula which Clairaut gave on his page 226. I have discussed the point in Art. 328. D'Alembert admits on his page 82 that Clairaut's more general formula on page 217 would supply all that was needed. D'Alembert quotes in his own favour, with respect to his controversy with Boscovich's translator, a passage from a letter to himself, written as he says, by one of the greatest geometers of Europe : see his page 83. 592. The next memoir in the sixth volume of D'Alembert's Opuscules Matlieinatiques is entitled Sur Veffet de la pesanteur au sommet et aupied des Montagues and more briefly Sur I attraction des Montagues ; this occupies pages 85... 92: it is followed by an Addition d V Article precedent on pages 93... 98. 593. A certain observer had reported that on the summit of a mountain in the Alps, 1085 toises high, a seconds pendulum had gained 28 minutes in two months ; so that gravity appeared to be greater at the summit of the mountain than at its base. D'Alembert proposes to shew how the fact may be explained, assuming the observation to be accurate. D'Alembert investigates the attractions of mountains of various shapes. The investigations are simple and satisfactory. In one case he supposes the mountain to be cylindrical, its height being small compared with the radius ; he obtains a result which was first given by Bouguer, and has since passed into the elementary books : see Art. 363. D'Alembert also investigates the influence exerted on a pen- dulum when it is placed in a valley between two mountains. If p be the mean density of the Earth, and p f that of the mountain, D'Alembert finds that supposing we accept the obser- vation on the Alps as trustworthy we must have p = ~ . This we o should now consider to be quite inadmissible, and so we should have no faith in the observation. But at the date of the memoir 383 the state of knowledge was different ; and D'Alembert says on his pa-vs 90, 91 : ...cette hypothese n'a rien de force; puisqu'on pent tres bien supposer que la densite moyenne de la Terre est moindre que la densite des couches qui sont a sa suiface. The words are hardly fair ; for the formula would make the mean density of the Earth scarcely one-third of that of the moun- tain. D'Alembert refers on his page 92 to Bouguer's work on the Figure of the Earth, pages 357 and following. D'Alembert says : On y trouve une Theorie de I'Attractlon des Montagnes, mais beau- coup moms generale que celle qui a ete 1'objet de ce Memoire. 594. On his page 93 D'Alembert refers to new observations with which he had become acquainted long after he had finished the preceding memoir. These observations seemed to shew that in a certain district of the Alps, attraction in ascending the moun- tains varied directly (not inversely) as the square of the distance from the centre of the Earth. He traces the consequence of this hypothesis. Let h be the height of the mountain, p' its mean density, p the mean density of the Earth, r its radius. Then by the investiga- tion referred to in Arts. 363 and 593 it appears that the attraction 47rpr 3 at the top of the mountain is =-7- ^r^ + 2?rp'/i, that is approxi- 3(r+ fi) mately / -f 2?r^ ( p -- } . If the attraction varies directly as 3 \ o / the square of the distance from the Earth's centre this must be 4-Trpr fr 4- A\ 2 , . . 4>7rpr f 2h\ equal to I - - ) , that is approximately to Q ( 1 H 1 . 3 \ t* / o \ i* / Hence we have this leads to p = -~ . o 384 D'ALEMBERT. The coincidence of this result with that in Art. 593 is certainly curious ; because it is a theoretical inference from observations which do not seem to have been influenced by theory. However there can be, I presume, no doubt that the observations must have been erroneous. Frisi alludes to the-matter ; see his Cosmo- graphia. Vol. 11. page 142 : he seems to treat the observations as fictitious. He says: JSTotitiis eiiim couquisitis undique accepi alpina ilia experimenta... omniuo esse supposita, et circa differentiam attractionum in vertice, et ad pedes montium Bouguerii tantum experimenta superesse qua? in investigationibus figure terrestris locum aliquem semper habere debeant. See also La Lande's Bibliograpliie Astronomique, page 532. 595. The next memoir in the sixth volume of D'Alembert's Opuscules Mathematiques is entitled Suite des Recherclies sur la Figure de la Terre; this occupies pages 99... 133; it is followed by some Remarques sur le Memoir e precedent on pages 134... 160. 596. The problem discussed is one which D'Alembert briefly noticed on pages 63... 67 of the volume : a homogeneous mass of fluid in the form of an ellipsoid of revolution rotates with uniform angular velocity round its axis of figure, and is supposed to be in relative equilibrium under its own attraction and the attraction of a distant body situated on the prolongation of the axis of figure; then the condition for this relative equilibrium is found and dis- cussed. Although the problem cannot be considered to be of any physical importance yet the analytical processes are both inter- esting and instructive. Let M denote the mass of the distant body, h its distance from the centre of the ellipsoid ; the axis of revolution of the ellipsoid when produced passes through M: take this for the axis of a. Then the distant body exerts an action T* a ^ * ne centre of the ellipsoid ; and then in the usual way we find that what we may call the disturbing action of the distant body at a point (x, y] is , My f equivalent to -p and -rjr parallel to the axes of x and y re- D'ALEMBERT. 385 spectively; the former in the direction in which x increases, the latter contrary to the direction in which y increases. M D'Alembert says nothing about the force j*\ we must in fact imagine it to be counteracted by an equal force applied at every point. Let us suppose that the equatorial axis of the ellipsoid is m times the polar axis ; and let k = *J(m 2 1). Suppose the density of the ellipsoid to be unity : then taking it to be an oblatum the attractions at (x, y] parallel to the axes of x and y respectively are by Art. 581 ~ (tf + 1) (k - tan-Vc) x and {(fc 2 + 1) tan" 1 A; - k} y. We have also the centrifugal force ofy parallel to the axis of y, where CD is the angular velocity. Hence putting X and Y for the whole forces at (x, y) parallel to the axes of x and y respectively, and estimating these forces inwards, we have Z= ~ (fc 2 + 1) (k - tan- k)*- Now we may apply Huygens's principle to obtain the condition of relative equilibrium. Thus X and Y must be positive, suppos- ing x and y to be positive ; and Xdx + Ydy = 0, must coincide with the differential equation to the ellipse which generates the ellipsoid, that is with xdx + prr-j = 0- Hence we obtain (# + !}(* -tan" 4}-?^ = (* + 1) [ J {(If + 1) tan' 1 k-k} + %- ?] ; and simplifying we have <J M S + tan-^-Sfe M T. M. A. 25 I I Mix ir -%*->. , 386 D'ALEMBERT. This is the fundamental equation of the problem; it agrees with D'Alembert's on his page 100, though with rather different notation. We shall, as in Art. 581, put <j>(fy for k* 597. We have hitherto supposed the ellipsoid of revolution to be an oblatum. If it be an oblongum our fundamental equa- tion still holds, only r as k = *J(m* 1.}, and m is now less than unity, <f>(k) contains impossible quantities which must be trans- formed. We have ,, 7N 3 + k z tan' 1 k 3 2 + m 2 tan'VO 2 -!) 3 $(/c) -- k k* m 2 -!' VK-1) m'-l" If m is less than 1. we find that - ^ ^ ^ - transforms V(w 1) 1 , l + V(l-m 2 ) the usual way into a ,,., -- 5x log ^ -- ^- = . 2l-m 3 l-l-w n 598. Our fundamental equation may be written thus O) 2 M We have to consider whether a value or values of m between zero and infinity can be found to satisfy this equation. Moreover, if m is less than unity, we must consider that the proper form for $V( m2 1)> f ree from impossible expressions, is we will denote this by ^ (w). That we have obtained the right equation for the case in which w is less- than unity, may be verified by an independent investigation of the attraction of an oblongum on a particle at its surface. D'Alembert himself indicates this method of confirming the result obtained by the ordinary use of Imaginary symbols : see his pages 134, 135. D'ALEMBERT. 387 599. Let us first consider the range of values of <f> (k), as k increases from zero to infinity. 4& 2 When k is very small $ (k) is approximately equal to , as JLo may be easily shewn by expansion. And </> (&) obviously vanishes when k is infinite. D'Alembert wishes to shew that <j> (k) is always positive ; see his pages 102 and 103. His demonstration is unsound. He shews that tan" 1 k - ^ is positive when k is infinitesimal ; and he k shews that this expression is positive when - -- j-, 2 has its greatest 1 ~f~ 3 fa value, namely, when k = V3. It is easy then to see that the ex- pression must be positive when k is greater than ^3. But it does not necessarily follow that as k changes from to \/3 the expres- sion is always positive. We may proceed thus. Put u = (3 -f & 2 ) tan' 1 k 3k ; then = 2k tan' 1 k - 3~p = 2 tan ( e - sin e cos &)> if t^" 1 k = e - Thus -57 is positive while k changes from zero to infinity ; and so u continually increases with k and never vanishes. Since <f> (k) is always positive and vanishes both when k is zero and when k is infinite, it follows that <' '(k) must vanish, once at least, within this range of values of k. We have moreover shewn in Art. 586 that <t>'(Jc) can vanish only once. We may observe that D'Alembert draws his diagrams consistently with the fact that <j> (k) vanishes only once, though as we have remarked he did not demonstrate this. 600. D'Alembert shews that ^fr(m) is always negative if m lies between and 1. We have, in fact, to shew that is always negative. D'Alembert's method is rather laborious : see 252 388 D'ALEMBEBT. his page 104. The best way is to expand in powers of \/(l Put t for V(l m2 ) > then we have 3-f, 1+* 3 Expanding the logarithm we find that (t* 2 4 0+-6T + - + <2n Thus as m increases from zero to unity, we have ^ (m) always negative, and numerically continually decreasing from infinity to zero. This continual decrease is not mentioned by D'Alembert, though he draws his diagram consistently with it. It will be convenient to give also the expansion of <>(&). L/I\ i i We have < k = l + L/I\ f < (k) = ( 3\tan*& 3 expand tan" 1 A;; thus we get 4P 8k* " 375 " 577 + ' ' ' (2n + 1) (27i + 3) Since Jc z m 2 1 = ?, we see by comparing these two ex- pansions that the value of < [*J(ni 2 1)} suffers no discontinuity as m passes through the value unity. This of course might have been held probable, but now it is demonstrated. The series for ^ (m) and <j> (k) furnish us with an expansion for <f> [*J(m* 1)}, which will remain convergent for values of m between and V2, the former extreme value being excluded. 601. Suppose we put M =0 in the fundamental equation of Art. 596 ; then we see that the equation cannot be solved by a value of m less than unity ; for the left-hand member would be positive, and by Art. 600 the right-hand member would be negative. Hence a mass of rotating fluid cannot be in relative equilibrium if it is in the form of an oblongum, the axis of rotation coinciding with the axis of figure. D'Alembert does not draw this inference from his formula. The theorem was first given by Laplace in his Thorie...de la Figure des Planetes, page 128. DALEMBERT. 389 602. From Arts. 599 and 600 we have the following results as to the value of $ [>J(m* . 1)}- When m increases from zero to infinity, </>{V(w 2 - 1)} begins by being negative infinity, increases algebraically, is zero when m = 1, then becomes positive and in- creases to a maximum, and finally reduces to zero. In the diagram we take m as the abscissa, and < {^(m z 1)} as the ordinate of the curve, and we consider ordinates positive when they are above the straight line OM: D'Alembert reverses this arrangement. o -Jf 603. Next we may proceed to consider the curve, the ordi^ nate of which is formed by adding to the corresponding ordinate of the preceding curve the term -p a * as required by the funda- f JT/L Tfy mental equation of Art. 598. M so that the fundamental equation becomes M When m is indefinitely small, f(m) is positive and indefinitely great ; when m is infinite f(m) vanishes. Let y denote an ordi- nate corresponding to the abscissa m ; then the curve determined by y=f(m) may take various forms. D'Alembert discusses the fundamental equation with great 390 detail, considering various cases which arise according to the values of - -- 3 and the different forms of the curve y=f(m). We will notice briefly some of the more interesting points which occur. Let us consider some of the peculiarities of the curve y =f(m). (1) Let m t denote the value of m for which <p {^(m" 1)) has its maximum value. If r 3 is less than < {*J(m* 1)} + n 2 ITflf J ff ft TH/+ we have/ (m) greater when e m m^ than when m = 1. And/(w) is greater when mm^ than when m = GO . Thus/(ni) must have some maximum value between m = 1 and m = oo . D'Alembert, pages 107 and 148. (2) It is possible that f(m) should be negative for part of the range between ra = and w=l. For this merely requires that 3 2 -f i^ (m) should be negative, or that -y^ + m z ^r (m) M should be negative. Therefore, if r- 3 is less than the numerically greatest value of mSjr (m), which is always negative between m = and m=l, there will be negative values of f(m). As m 2 i/r(m) vanishes when m = and when m = 1, there will be a numerically greatest value of m within this range. D'Alembert, pages 111 and 148. (3) If, however, r- s is greater than the numerically greatest irri value of mS/r(ra) within the range from m = to m = 1, then f(m) is always positive from m to m = oo . (4) It is possible to have such a value for 73 that f(m) shall decrease continually from m = to m oo ; that is, f (m) shall be always negative. D'Alembert, pages 117 and 120. First, from m = to ra = l. Here we have (8 + m 2 )m . 2 (1 -m 2 )* g 1 ~ VO - ) m (1 - m 2 ) D'ALEMBERT. 391 This will be negative within the range, if algebraically 2^r. (7m 2 +2)m 2 (S + m^m*, 1 + V(l - 2 ) rr is greater than ~. ^ -- - - r log- -- -^ -- ^ . irtf (1-my 2(l-m 2 ) 1 *1-V(1-) The expression on the right-hand side vanishes when m ; Q and by evaluation it will be found to be when m = 1. It is 15 always finite between these limiting values ; and if -7-3 is greater TT/l than the algebraically greatest bf the values, f (m) will be nega- tive from m = to m = 1. Next from m = 1 to m = <x> . Here we have . tll .dk 2M (7^ + 9 9-ftf. _ 17 )m 2M f (m) = <b [le] -, --- r^ i = { 7 <, /^ r^r -- 7i tan Ur T -- 71 < > ^ ^ ' dm 7rk 3 m* kI + k* & 4 A: 9 * where 7c 2 = m 2 - 1. This will be negative between k = and k = x , if algebraically 2J/ . (& 2 -f I) 2 f 7fc 2 + 9 9 + k\ _! 7 ] rf 1S ^ reater than -nT" {FW+I)" ~^" Q The expression on the right-hand side will be found to be - lt> when A; = 0, as it should be from above ; and it is negative infinity when Jc = oo . Hence there must be a greatest value among the positive values which it can take. If ^ is greater than this value, f (m) will be negative from m == 1 to m = oo . 9 1/" If then --73 be greater than the greatest of the two values TTfi' which have thus presented themselves, f'(m) will be negative from m = to m oo . g 604. The numerical result y-^- which occurs in the preceding Article may be easily verified. In fact, it is the value of i|r'(m) when 7?i = 1, or of ,<(&) which is required* Take the latter; then we have <f>'(k) ^ , that is, <f>' (k) y , that is, by Art.- 600,- 392 D'ALEMBERT. m (8 32& \ ~k V 15 S^~ " ' " J ' a wnen m = 1 so that = 0, this becomes o =-= . The same result will follow by the aid of Art. 600 from the 15 value of ^r'(w). 605. D'Alembert shews that the problem may in certain cases have two or three solutions for given values of G>, M t and h. He makes some remarks as to what we should now call the stability of the relative equilibrium, like the remarks on pages 56 and 57 of the volume which we have noticed in Art. 582. See his pages 112.. .115, 126. ..128, 153. 606. In the fundamental equation of Art. 598 put m 1 ; then since <f> }\/(ra 2 1)} =0 when ra = l, we have ^ = SM 2ir ~ 2?r/i 3 ' Hence this relation must hold in order that a sphere may be a possible form of relative equilibrium. 607. When we have obtained a solution of the fundamental equation, it will still be necessary to advert to the condition stated in Art. 596, that X and F must be positive if x and y are, before we can say that relative equilibrium exists. It will be sufficient to ensure that one of them is positive, because if the fundamental equation is satisfied, we know that X and F are of the same sign, supposing x and y to be. D'Alembert pays proper attention to this point : see his pages 105, 116, 117, 122, 123. Let us, for instance, consider the value of F. Hence we see (tf + lJtaif^-A; o> 2 M that we must have * p greater than ^ -- 9~P* Denote the former expression by v ; then it will be found that (fo _ 3fc - (3 + 2 ) tan" 1 ^ dk~ ~W~ By Art. 599 we see that -^r is always negative for real values of k ; and so for such values v is greatest when k = : and then 2 D'ALEMBERT. 393 When we put V(w 2 -l) for Jc, and suppose m less than 1, we get dv dv dk m , tan" 1 2V(l-m'0 By Art. 600 we know that this is always negative if m lies between and 1 ; and so for such values v is greatest when m=0. 1 + V(l - 2 ) 1 - log .. ,/-. *f + ^ 5> so that when 2 )i 1 y(l m) 1 m But v = - m = we have v = I. Thus as m varies from zero to infinity, v continually dimin- ishes from unity to zero. See D'Alembert's pages 116, 117, 151, 152. The fact that v continually diminishes as m increases may also be shewn by putting the value of - - thus : dv m , f ,, , ,^ ~~ */J 5 this is always negative, for the factor ${\/(m 2 1)} is negative when m is less than 1, and the factor = ^ % is negative when m is greater than 1. It follows from this discussion that there can be no relative equilibrium if -^ -- , 3 is algebraically greater than unity. See D'Alembert's page 117. 608. Now let us consider the value of X. Hence we see -) M ,,. that we must have - ?3 ' ^ - greater than r 3 . This * ?3 K leads us to investigate the greatest value of the former expression. TA -111, r ^^ It will be found that this expression = 1 ^ = 1 v ; and as v continually diminishes from unity to zero, this expres- sion continually increases from zero to unity. It follows that K/T there can be no relative equilibrium if ' is greater than unity. ATT/1 See D'Alembert's page 124. 394 D'ALEMBERT. 609. D'Alembert suggests another mode of obtaining solu- tions of the problem: see his pages 128... 132. Let m be an abscissa and y an ordinate as before; and let k = */(m* 1). Then draw the curves ^(fc-tan- 1 ^) M ' (fc'+IJtaif 1 *: 1 M and ~- - ~ At a point of intersection of these curves the corresponding value of m will satisfy the fundamental equation; and if the value of y at the point of intersection is positive, the resultant force at the surface tends inwards: therefore with the value of m thus obtained relative equilibrium will subsist. It is sufficient by Art. 608 to confine ourselves to the case in which -; r- 8 is less than unity. In drawing the curves the results obtained in Art. 607 will be found useful. Thus, for instance, the equation to the first curve may be written and we know that v diminishes continually from unity to zero as m increases from zero to infinity. Hence y begins by being nega- tive infinity, vanishes and changes sign once and only once, and is zero when m is infinite. When m = 1 we have y = 2 n - = - o ; this is posi- TTfl O O TT/V M 1 tive or negative according as -5 TS i g ^ ess or greater than - . 610. Instead of the two curves of the preceding Article, D'Alembert suggests in his pages 158... 160, that we may take the two curves M and D'ALEMBERT. 395 611. D'Alembert discusses at some length two analytical matters which present themselves. On pages 134... 142 he treats of difficulties which may occur in the use of the symbol J( 1). For example, suppose we require the product of *J(-a) into V( &) ^ n one hand we may take for it *J(-axb), that is, *J(al>). On the other hand we may take for it *J(a) x V( 1) x VW x V( 1)> tnat ^ J(ab) x V(- 1) x V(~ 1)> that is, - J(db). On pages 142. ..145 he shews in various ways that a; log a; is zero when x is ; and so also is 'y? log x q where p and q are positive and finite. 612. The next memoir in the sixth volume of D'Alembert's Opuscules Mathe'matiques is also entitled Suite des Recherches sur la Figure de la Terre; this is a continuation of the preceding memoir; it occupies pages 161... 197: it is followed by some Eemarques sur le Memoir e precedent on pages 19 8... 2 10. 613. In the preceding memoir D'Alembert had considered the relative equilibrium of a mass of rotating fluid in the form of an ellipsoid of revolution acted on by the disturbing force of a distant body, situated on the axis of rotation produced. In the present memoir he generalises the problem by giving any situation to the distant body, and by taking for the fluid mass the form of an ellipsoid, not necessarily of Devolution. 614. We shall use notation more symmetrical than D'Alembert's. Suppose then that the fluid is in the form of an ellipsoid. Take the axes of x, y, z to coincide with the axes of the ellipsoid ; let 2a, 2&, 2c be the corresponding lengths of the axes. Let there be a distant body of mass M\ and let its co-ordinates be I, m, n respectively : put ft* = 1? + m 9 + n*. Suppose the fluid to rotate with angular velocity &> round an axis, the direction cosines of which are X, /*, v. We have to farm the conditions for relative equilibrium. Now here we must observe that the distant body must, in fact, be supposed to share in this rotation of the fluid mass. D'Alembert never notices this fact, though it is really involved in his process. In the particular case of the preceding memoir, in which the 396 D'ALEMBERT. distant body is supposed to be on the axis of rotation, we may practically regard the distant body as fixed ; but we cannot in the present memoir. A particular case of the present memoir, as we shall see, was afterwards discussed by Laplace ; in this case the Moon is taken to be the fluid mass, and the Earth to be the distant body. See Laplace's TMorie...de la Figure des Planetes, pages 113... 116. 615. Let P be any point of the fluid ; let x, y, z be the co- ordinates of P. The attraction of the fluid ellipsoid parallel to the axes of x, y, z respectively will be Ax, By, Cz respectively where A, B, G are certain constants. D'Alembert in effect briefly states that this can be easily shewn in the way in which Maclaurin treated the attraction of an ellipsoid of revolution ; this is true, and it is to be noted that we have here, for the first time, the important extension of Maclaurin's result from an ellipsoid of revolution to the general ellipsoid. See D'Alembert's page 165. But as we shall hereafter point out, Frisi had previously gone some way in this direction : see his De Gravitate, pages 157 and 159. 616. The attraction of the distant body at P parallel to the Mil x} axis of x is - - - ' ^ ; the disturbing part of {(j_ a ). + ( TO _ y )- + ( n _a)}i' this is approximately MX SMI (Ix *f my 4- nz) "W* Jff MX , . , 7 say ^r -i ~ where u is put for Ix + my + nz. It is only the disturbing part of the action of M which D'Alembert regards; he makes no allusion to the other part, that is, -^3 in this case. See Art. 596. Let denote the centre of the ellipsoid ; let Q denote the foot of the perpendicular from P on the axis of rotation ; then the so-called centrifugal force is ca?PQ, and we require the re- solved part of this. We have to project PQ on the axis of x and by a known theorem of projections we may take the difference of the projections of OP and OQ for the projection of PQ. D'ALEMBERT. 397 flfl Thus we obtain w 2 (OP. -T OQ cosX) ; and this = G> 2 (a; - OQ cos X) = &>* (a - OP cos P0<? cos X) = o> 2 {# (x cos X + y cos //, + 2 cos z^) cos X} = a) 2 (03 v cos X) where v is put for x cos X + y cos /z. + 2 cos v. Let X denote the whole force parallel to the axis of x, esti- mated inwards ; then MX SMlu 2 , >x X=Ax + -^--j^-a) *(x-v cosX). Similar expressions hold for the attractions parallel to the other axes, which we will denote by Y and Z respectively. D'Alembert's method is substantially equivalent to this though his notation is less symmetrical. 617. The conditions for relative equilibrium are Y x -Y y - 7 - Z -/X ~"J~ o J- ~!~ ~T* - J ~~?~ 9 a 2 V c 2 Take the equation Xcfy = Yb*x ; this must be identically true, and so we may equate the coefficients of xy, x*, y*, xz, yz. By equating the coefficients of xy we obtain 2 f. M 3MP , + ~ ^ " ) f w , M = ( + ^ -- Tt By equating the coefficients of # 2 , and by equating the coeffi- cients of 7/ 2 , we arrive at the same condition, namely, SMlm -- rjj -- h G) cos X cos p 0. By equating the coefficients of xz we have SMmn 2 -- gr f- CD cos a cos v = 0. ^t By equating the coefficients of yz we have SMnl D5 cos ^ cos X = 0. 398 D'ALEMBERT. In like manner we may take the equation Xcfz = Zc*x ; by so doing we shall find that we get only one new condition. The whole results may be written thus: _ tjir.tlllt n *JJ.Ulllll> n .- - = ft) 2 cos /j, cos v, = co cos v cos X, jTg = &) cos X cos ytt. ./I ll Jt M SMI 2 2 . 2 \ ,,/ , JW 31/m 2 2 . 2 \ j^ - -g 5 w 2 sin 2 X 1 = 6 2 f B + -^3 -^5 a) 2 sin 2 /^ 1 .-X- :jf M 3Mn* 2 . 2 \ F " ~^~ "" sm r j ' 618. As a particular case of the preceding investigation, sup- pose that there is no distant disturbing body ; then M = ; thus cos /J, cos v = 0, cos v cos X = 0, cos X cos fjL 0. Hence two of the three cosines cos X, cos p, cos v must vanish ; so that the rotation must be round one of the principal axes of the ellipsoid. Hence we see that the case taken in Jacobi's theorem is the only case in which an ellipsoid of fluid rotating round a diameter can remain in relative equilibrium. A statement which has been recently made to the contrary by Dahlander and by Schell is inaccurate : see the Proceedings of the Royal Society, Vol. xxi. 619. Return to the conditions obtained in Art. 617. Let us suppose that Z, ra, and n are not zero. The first and the second of these conditions give ra _ cos fj, ^ I cos X ' the second and the third give n cos v m cos /A Hence the radius vector to the distant body coincides in direc- tion with the axis of rotation ; thus I m n and then from any of the first three conditions we get 33f ^ =G); and the other conditions reduce to D'ALEMBERT. 399 these last will be satisfied if a = b = c, that is if the fluid mass be spherical. The particular case in which the radius vector to the distant body and the axis of rotation coincide in direction presents itself in D'Alembert's memoir ; but he does not pay much atten- tion to it : see his page 200. He also notices a particular case in which it is given that two of the three a, b, c are nearly equal : see his page 209. But he does not notice that we may have a sphere exactly if I m n , 3M and to = - nr . cos X cos p cos v 620. It will be interesting to enquire if the conditions in the preceding Article can be satisfied in any other way besides having a = b = c ; this enquiry leads us a little beyond the point at which the theory of the attraction of ellipsoids had arrived at this date. Let V denote the mass of the ellipsoid ; then we know that x*dx 3FT 1 a J This result was given by Laplace in his Thforie ...dela Figure des Planetes, page 92 ; as we shall see D'Alembert himself first obtained it but rejected it in the seventh volume of his Opuscules Mathe'matiques. Assume x = 7-^ : ; then we find that ."2 Jo (tf + s)D' where D stands for V{( a * + 5 ) (^ + 5 ) ( c * In like manner we have Put </>* for -7^-; then the conditions we have to examine may be written 400 D'ALEMBERT. Jo Jo 2 o hence we see that these conditions cannot be satisfied if a, b, c are all unequal ; for they would lead to two different values of $' 2 . But suppose two of the three, a, b, c to be equal ; say a and b : then our conditions reduce to sds __3F = 2 Jo and this is quite admissible if b, c, Fand < be properly adjusted, whether b is greater or less than c. 621. If ?, ra, n are all different from zero we have the case discussed in the preceding two Articles, in which the radius vector to the distant body and the axis of rotation coincide in direction. D'Alembert himself pays little attention to this case : indeed in his page 200 he seems to consider that it cannot occur. Let us now return to the general conditions of Art. 619 ; and suppose that I, m, n are not all different from zero. Suppose for example that n = 0; then it follows from the first and second conditions that either cos v = 0, or else cos X = 0, and cos //. = : if we suppose the latter, then I or m must also = 0. In the former case, the axis of rotation is in the principal plane corresponding to a and b ; in the latter case the axis of rotation coincides with the axis corre- sponding to c. In each case the axis of rotation and the radius vector to the distant body are both in one of the principal planes of the fluid mass. 622. In Arts. 619 and 620 we see that the supposed ellipsoid is either a sphere or an ellipsoid of revolution; and in Art. 621 we see that the axis of rotation and the radius vector to the distant body must, be in one of the principal planes of the fluid mass. Combining these two results, we may say that in every case in which the relative equilibrium is possible the axis of rotation and the radius vector to the distant body must be in one of the princi- pal planes of the fluid mass. D'Alembert arrives at this result, D'ALEMBERT. 401 and confirms it by some general reasoning which is not very cogent: see his pages 198... 200. 623. As a particular case of Art. 617 let us suppose we have given that the axis of rotation and the radius vector to the distant body are at right angles. This may be considered to hold with respect to the moon supposed fluid, the distant body being the Earth. Since here we have not the case of Arts. 619 and 620, it follows that one or two of the three I, m, n must be zero. Sup- pose n = ; then from the first two conditions of Art. 617, we shall find either cos v = 0, or both cos X = 0, and cos //, 0. I. Suppose cos v = 0. Then the third condition is , ^ 3 ft) COS A, COS fJb = Now by our hypothesis that the two directions are at right angles, this would give o> 2 = ~g- , if we suppose that Im does not vanish ; this is impossible. Therefore Im vanishes. Hence we must have either I = 0, and cos ^ = 0, or m = 0, and cos A. = 0. II. Suppose cos X = 0, and cos /* = 0. Then the third condition shews that Im = 0. Therefore either I = 0, or m = 0. Hence we must have the axis of rotation coinciding with one of the principal axes of the body, and the radius vector to the distant body coinciding with another. The result might have been anticipated perhaps ; and we shall find that Laplace assumes it as evident : see the reference in Art. 614. 624. We have seen in Art. 622 that the axis of rotation and the radius vector to M must always be in one of the principal planes of the ellipsoid. We will suppose that n 0, and cos v = 0. Hence the conditions of Art. 617, reduce to SMlm T* = ca> cos X cos /*, T. M. A. 26 402 D'ALEMBERT. 2 f , M 3JfZ 2 ) a 2 14 + -jp - -gg w 2 sin 2 Xj And in virtue of our supposition that 71 and cos v vanish we have cos 2 X + cos 2 /* = 1, Z 2 -f m 2 = jK 2 . As to whether these equations are consistent nothing is said by D'Alembert; we have discussed one case of the general problem in Arts. 619 and 620, but the matter is not of sufficient importance to detain us longer. 625. D'Alembert begins on his page 174 an investigation of the attraction of an ellipsoid on any particle at the surface. This amounts to seeking the values of the A, B, C of Art. 615. He makes some simple and useful remarks on his pages 174. . .176; we will give an example of them. Suppose the semiaxes of an .ellipsoid to be r, r (1 + a), and r (1 + &), where a and /3 are very small. Let the approximate value of the attraction be required for a particle situated at the end of the semiaxis r. We may assume that this attraction will be ^- (1 + pai+q/3), where p and q are certain constants to be determined : this assumption depends on the fact that if a and ft vanish, the body becomes a sphere, and the attraction then is -= . Next we may admit that p = q\ be- o cause the attraction ought to remain unchanged if we interchange the second and third semiaxes. Hence the attraction becomes ^ {1 +p (a + f$)}. Now we can determine p. For if we suppose a = /3 the ellipsoid becomes an ellipsoid of revolution, and the attraction of such a solid on a particle at the pole is known : hence equating this known attraction, estimated approximately, to (1 + 2pa) we determine p. We should thus get p = . 626. D'Alembert attempted to find the attraction of an ellip- soid by decomposing it into slices in various ways ; but he does not succeed in effecting the integrations. We know now that the D'ALEMBERT. 403 result can be expressed by means of elliptic integrals, but not by circular arcs or logarithms. We will briefly state the methods of decomposition of the ellipsoid which he tries. The attracted particle is supposed to be at the end of the semiaxis c. I. Suppose a plane to pass through the attracted particle, and also through the tangent to the ellipsoid at that point which is parallel to the axis 2a. Let this plane turn round the tangent line and cut the ellipsoid into wedge-shaped slices : see D'Alembert's page 180, This decomposition is like that used by Thomas Simpson ; which we have noticed in Art. 279. II. Instead of using the tangent parallel to the axis 2a, we may use the tangent parallel to the axis 2b. III. Suppose a plane to pass through the axis 2c and to turn round, and thus cut the ellipsoid into wedge-shaped slices : see D'Alembert's page 183. This decomposition is like that used by Maclaurin ; which we have noticed in Art. 255. IV. Or the ellipsoid may be cut into laminae by a plane which is always at right angles to the axis 2c : see D'Alembert's page 184. 627. For the case of an ellipsoid in which two of the axes are very nearly equal D'Alembert obtains approximate values of the attraction at the end of the principal axes : see his page 192. A mistake in the results is corrected on page 424. The approximate results just referred to are applied by D'Alembert to the question of relative equilibrium which was proposed at the beginning of the memoir : see his pages 194... 197. He finishes in a patronising tone : Je ne doute point que cette nouvelle Recherche ne donnat lieu a plusieurs remarques curieuses ; mais je les abandonne a d'autres Ge"o- metres, la matiere n'ayant plus aucune difficult^. 628. The next memoir in the sixth volume of D'Alembert's Opuscules Mathematiques is also entitled Suite des Recherches sur la Figure de la Terre; this is a continuation of the preced- ing memoir; it occupies pages 211... 246 : it is followed by some Remarques sur le Mtmoire precedent on pages 247... 25 9. 262 404 D'ALEMBERT. 629. D'Alembert now proposes to extend the problem of the preceding memoir by supposing several distant attracting bodies instead of the single distant attracting body there considered. This extension becomes very easy with the aid of modern sym- metrical notation. Let M v M y M. A ... denote the masses of the various distant bodies respectively ; let 1 19 m v n^ be the coordinates of the first body, R^ its distance ; and let similar notation hold with respect to the other bodies. Then instead of the first equation of Art. 617, namely SMmn k = w cos /* cos v > n ** SMm.n. . SMjnji.. 3Mjn..n. we now have ?nr^ -f frr- 1 H -- Snr 3 -f- . . . = o> HI M % M a ., ,, m* Mmn which we may write thus 32 = a> cos //, cos v. cos ^ cos v, similarly the other equations may be expressed. D'Alembert himself does not proceed in this way nor adopt this notation. He uses spherical trigonometry. It may be observed that he demonstrates the expression for the cosine of an angle of a spherical triangle in terms of the sines and cosines of the sides ; he starts from formulae for a right-angled spherical triangle which he assumes : see his pages 247 and 248. As we have remarked in Art. 614, the distant bodies must be supposed to rotate with the fluid mass ; though D'Alembert does not notice this fact. And as in Arts. 596 and 616, D'Alembert says nothing about certain forces which are not what I have called disturbing forces. 630. The only point which appears to be of any interest in the problem is a remark which D'Alembert makes on his page 253 ; the remark amounts to this : if the axis of rotation and the radii vectores to the distant attracting bodies are all in one plane that plane must be a principal plane of the ellipsoid. He does not de- monstrate this, but seems to rely on the principle of symmetry as in the corresponding theorem for a single distant attracting body : see Art. 622. We will examine the theorem. Suppose that the equation to the plane is ax + fiy + yz = ; so that D'ALEMBEKT. 405 a cos \ + cos fi + 7 cos v = 0, and so on. Take the three equations 32 ~- = o) f cos /-t cos v, ^~7j*~ = ^ cos v cos ^ -^ Mlm 32 -r = ft) cos X cos a. L Substitute in the first of these for n v n a , ..,, and for cos v ; n ^ 2 , N thus 32 - ps - = w cos fj,(a.co$\ + @ cos /*) therefore by means of the third equation we obtain 0<l ifra 2 2 2 32 -=g- = ft) z COS* ylt. Similarly 32 -7*- = ft) 2 cos 2 X, 32 -^ v? cos* v. Hence the first of the three equations becomes Mmn /^ Mm* ~ Mn z . (M.m.n. Mjnji- Mjnji. Squarmg we get j-^-H + -^p+ -^+ ... j _* Mjn'Mjn,' r'^r ^? v This by common Algebra leads to 1 = ? = 3 = . . W l ^2 ^3 In this way we see that all the radii vectores to the distant "bodies must coincide. Thus the case reduces to that of Art. 619. But suppose, as in fact D'Alembert does, that the plane in which the axis of rotation and the radii vectores to the distant bodies lie is perpendicular to a principal plane ; let its equation be 406 D'ALEMBERT. Then as before we can obtain from our three equations, but we do not now have also 3X - = ft) 2 cos 2 v. The equations which correspond to the last two of Art. 617 are C 2 a 2 N for a) 2 sm* y = w 2 cos 2 X + cos 2 = -^ If D'Alembert's remark were universally true the equations connecting a, b, and c ought to be impossible, or inconsistent with the others, if a, 6, and c are unequal. But this does not seem to be the case. By the method of Art. 620, we get from them ^_2 J f il J2 8 2 ~M 37 and 2c S - = - and these if c 2 (a 9 + 6 2 ) 6V is positive present nothing impossible. As an example we might suppose two distant bodies, and take l t = BZ COS ^> m 2 = ^2 COS P* n 3 ^2 COS V ' Then it will be found that our first three equations give O TUT o> 2 = --prj ; and we have only to ascertain if this is consistent with the last two equations, the form of which has just been given. Thus we have to put (c* + s)sds D'ALEMBERT. 407 It will be found that these lead to values of ^f and =--J which **i a i are certainly positive if (a 2 c 2 ) (c 2 - 6 2 ) is positive; for then also c 2 (a 2 4- 6 2 ) - &V is positive. It is manifest that this condition may be satisfied ; and thus D'Alembert's remark is not true. 631. D'Alembert on his page 216, refers to Maclaurin's Essay on the Tides, as containing a little matter bearing on the problem discussed in this memoir; but Maclaurin had not effected much. Maclaurin did not shew that the figure of an ellipsoid would satisfy the conditions of equilibrium ; nor did he show how to determine the position of the axes of the ellipsoid. D'Alembert says of his own memoir: Nous avons de plus demontre' dans celui-ci que la figure du sphe'roide est elliptique... However he does not shew that the figure is an ellipsoid, but only that it may be an ellipsoid. 632. D'Alembert says on his page 217, that he will conclude with some detached reflexions bearing on the Figure of the Earth. 633. He says that among the solutions hitherto given of the problem the only one which is exact is that which supposes the spheroid to be fluid and homogeneous ; the other solutions being approximations. Suppose that a is a very small quantity; and we have found that neglecting a the equation of relative equili- brium is satisfied for a certain figure ; we must not say that this figure exactly satisfies the conditions of relative equilibrium. But D'Alembert suggests that if we give to the figure a certain small change of the order a 2 the conditions of relative equilibrium may be rigorously satisfied ; and he considers it a plausible supposition that there may be an infinity of figures in which the relative equilibrium will subsist rigorously : see his page 223. Probably few persons will agree with D'Alembert in considering this supposition plausible. 634. D'Alembert returns on his pages 225. . .230 and 254. . .259, to his favourite equation relating to the ellipticity of fluid sur- rounding a solid nucleus : see Arts. 376, 430, and 590. We shall briefly notice some points that arise. On his pages 227... 229, D'Alembert criticises as inexact certain formulae on page 247 of Clairaut's work, and thus as affording an insufficient proof of Clairaut's theorem which is founded on them. 408 D'ALEMBERT. But, as might be expected, D'Alembert is wrong and Ckiiraut is right. The fact amounts to this : what I have called for instance A in Art. 336, is called A by Clairaut. Now D'Alembert really supposes A to stand for an integral taken not from to r v but from some value say r a up to r x : and thus he wants to add terms to Clairaut's formulae. Plana rightly takes the side of Clairaut: see Astronomische Nachrichten, Vol. xxxvm, page 245. On his pages 254, 255, D'Alembert gives, without any prepara- tory statements what is really a more exact investigation of the problem of Art. 376. He thus arrives at the result which I have given in Art. 377,. in which the difference between r' and r l is not neglected. In this investigation however he assumes on the second line of his page 255 the expression for the force at right angles to the radius. In Clairaut's investigations the necessary results are demonstrated, D'Alembert does not observe that the theorem is included in a more general one which he had demonstrated like Clairaut : see Art. 443. In the formulas of Art. 376, suppose that e = e' ; then we get e = - < where < stands for w 2 -r ~ . This result is independent of p ; it is the same as we should get for a homogeneous fluid. D'Alembert seems to attach special importance to this result : see pages 79, 225, 256 of the Volume. But the result is what might be expected. Suppose a homogeneous fluid rotating in relative equilibrium : solidify all but a film of fluid ; the relative equili- brium will not be disturbed. If we consider the film so thin that its action on itself may be disregarded, it is kept in relative equi- librium by the attraction of the solid part. Hence if we alter the density of the fluid film, it will still be kept in relative equili- brium. 635. On his page 231 D'Alembert refers to the demonstration he had given of the proposition that an oblatum is the only form of relative equilibrium for a revolving fluid : see Art. 575. That demonstration we pronounced a failure. From what he now says, it appears to me that he overlooks the consideration brought forward in Art. 577, as to what his theorem would have esta- blished if the demonstration had been sound. D'ALEMBERT. 409 636. D'Alembert devotes his pages 232... 246 to investiga- tions relative to the attraction of an ellipsoid on an external particle. He confirms by analysis Maclaurin's proposition respect- ing the attraction of confocal ellipsoids of revolution on an exter- nal particle which is on the line of the axis or in the plane of the equator. But D'Alembert was unable to extend this as Maclaurin did, to the case of ellipsoids not of revolution. D'Alembert says on his pages 242 and 243. Je soupgonne done que M. Maclaurin s'est trompe dans 1' art. 653 de son Traite des Fluxions, quand il a dit que sa methode pour trouver Tattraction d'un spheroide de revolution dans le plan de 1'equateur, ou dans 1'axe, pouvoit s'appliquer a un solide qui ne seroit pas de revo- lution AU reste, ce n'est ici qu'un doute que je propose, n'ayant pas suffisamment examine la proposition de M. Maclaurin, qu'il se contente d'enoncer sans la demontrer. As we have stated in Art. 260 Maclaurin really demonstrated the theorem which D'Alembert considers to have been only enunciated, and the truth of which he here doubts. Subsequently, as we shall see, D'Alembert conquered his doubts and demon- strated the theorem : he was the first person who drew attention to the theorem and demonstrated it after Maclaurin himself. 637. The next memoir in the sixth volume of D'Alembert's Opuscules Mathtinatiques is entitled Sur les Atmospheres des Corps Celestes; it occupies pages 339... 359. 638. The first paragraph explains the object of the memoir : Le but des Recherches suivantes est de donner sur 1' Atmosphere des Planetes quelques Remarques que je crois nouvelles, et de corriger en meme-temps quelques meprises ou des Auteurs cele"bres sont tombes sur cette matiere. D'Alembert refers on his pages 345, 347, 349 and 350 to Mairan's Treatise on the Aurora Borealis; he refers to Euler on his page 350 ; and to Maupertuis on his page 358. Thus, I presume, these are the celebrated authors whose mistakes he proposes to correct. 639. D'Alembert obtains his fundamental equation in an un- satisfactory manner. He assumes that the stratum of the air in 410 D'ALEMBERT. contact with the surface of the planet is a level surface; then he takes an exterior level surface ; and he makes what he calls the weight of a column terminated at these surfaces constant. He ought not to assume that the surface of the planet is a level surface for the air. Suppose co the angular velocity, r the distance of a point in the atmosphere from the centre of the Earth, 6 the angle which r makes with the polar axis, M the mass of the Earth. Then by the usual equations for relative equilibrium I dp M Q 1 dp M . Q . -- = -- IT cos 0, ~ = -- r. sm 6 + cox. p dy r p dx r Hence the equation to a level surface is * M co*x* -\ ~- = constant. T A Let r v and r 2 be the values of r at the equator and the pole respectively in the same level surface ; then M <*\? _M V* 2 "V The matter is discussed by Laplace, as we shall see hereafter; but nothing is really added to what we find in D'Alembert's memoir. D'Alembert shews that the zodiacal light cannot be caused by the atmosphere of the Sun : the remark is repeated by Laplace. See the Mecanique Celeste, Livre III., Chapitre vn. 640. The form of the atmosphere is determined by a curve of which the equation in polar coordinates is It may happen that corresponding to a given value of 6 we have two positive values of r, and one negative value. The three values would all be regarded in tracing the curve according to modern notions. D'Alembert touches on the subject in his pages 347... 349. We may state that his opinion briefly amounts to rejecting the negative value of r entirely. He observes, in fact, that if we put >J(x* + #*) for r, and clear of radicals, we obtain an D'ALEMBERT. 411 equation of the sixth degree ; and this gives a branch correspond- ing to the negative value of r just mentioned. But according to him this new branch does not belong to us. However, he is not so much regarding the curve itself as the physical problem from which it arose. 641. Hitherto we have not supposed any action on the at- mosphere except that of the planet to which it belongs; but D'Alembert proceeds to consider the action of one or more other planets. As in the case of a revolving fluid, when he introduces a distant planet he first puts it on the prolongation of the axis of rotation : see his pages 354 and 355. Next he supposes the distant planet to have any position. As before too he really supposes the distant planet to preserve the same relative position, so that, in fact, the distant planet must be supposed to rotate with the planet which carries the atmosphere. See Art. 629. 642. The mode in which D'Alembert finds what we should now call the pressure at any point of the atmosphere, when there is besides the planet itself, a distant planet acting, may be noticed. See his pages 35 5... 357. We know that the polar equations for relative equilibrium are ~ = ~ Now, in fact, he only considers the first of these equations. The value of p found from this must give the right result, pro- vided we remember that the so-called arbitrary constant must be, if necessary, regarded as a function of 6. But without working out the problem fully in rectangular coordinates, we easily see that the value of p must be such that 6 never enters alone, but always accompanied by r. Thus p cannot contain any arbitrary function of 6 alone. Therefore, the first equation alone is sufficient for finding p. D'Alembert himself, however, gives no explanation of his process. 643. In the Nouveaux Memoires de VAcadtmie. . . of Berlin, for 1774, published in 1776, we have extracts from two letters ad- dressed by D'Alembert to Lagrange; see pages 308... 311 of the 412 D'ALEMBERT. volume. D'Alembert had discovered that Maclaurin's theorem, about which he formerly doubted, was really true ; and here he sends to Lagrange sketches of two demonstrations : see Art. 636. The demonstrations are given at full in the seventh volume of the Opuscules Maihtmatiques, to which we now proceed. 644. The seventh volume of D'Alembert's Opuscules Mathe- matiques was published in 1780 ; a memoir entitled Sur I attrac- tion des Sphe'roides EUiptiques, occupies pages 102... 207; this is followed by some Remarques sur le MJmoire prudent on pages 208... 233. From page 208 we learn that the Remarks were written long after the memoir ; and, therefore, the memoir must have been written long before 1780. D'Alembert says that his attention had been turned to the subject again by reading the excellent memoir by Lagrange in the Berlin Mtmoires for 1773. 645. The first part of the memoir, which occupies pages 103. ..116, is devoted to the proof of Maclaurin's theorem: see Art. 636. D'Alembert starts from formulae given in the sixth volume of his Opuscules Mathe'matiques ; and by three different methods arrives at the required result. One of these methods occupies D'Alembert's Article 30 ; it is curious from its obscurity. When carefully examined it is found to be equivalent to a circuitous method of arriving at the ex- pression Trab for the area of an ellipse, of which a and b are the semiaxes. D'Alembert, on his page 114, corrects an important misprint in Lagrange's memoir in the Berlin Memoires for 1773. 646. The second part of the memoir, which occupies pages 116... 159, is devoted to the discussion of two formulae relating to the attraction of an ellipsoid, which were given on pages 180 and 184 of the sixth volume of the Opuscules Mathematiques : see Art. 626. We will briefly indicate, by modern methods and notation, the nature of these formulae. Suppose we wish to find the attraction of an ellipsoid, the axes of which are 2a, 2&, 2c, on a particle at the end of the axis 2c. D'ALEMBERT. 413 We use the method I. of Art. 626. Taking this point as origin, we have for the equation to the ellipsoid, 2* Put x = r cos 0, y = r sin 6 cos </>, 2 = r sin sin < ; then 2 sin sin < _ /cos 2 sin 2 6 cos 2 </> sin 2 6 sin 2 <ft\ The attraction which we require is equal to 1 1 Idr sin d6 c?</> . sin 6 sin <, that is, 2a 2 5 2 c 1 1 TF* *~/\ g 9 . 0/1 X y . 2 A . 2 . . J J c cos -1- a c sin ^ cos <> + a 6 sin ^ sin 9 The limits for 6 are and TT ; the limits for <f> are ^ and ^ . 2* Zt Now suppose we integrate first with respect to 6 ; put t for cos 6 ; thus we obtain the form (1 - Q dt a 2 c 2 cos 2 <j) + a?b 2 sin 2 <f>-{-t* (6 2 c 2 a 2 c 2 cos 2 <f> a?b* sin 2 (f>) ' There is no difficulty in integrating this ; but the form of the integral is different according to the sign of 6 2 c 2 aV cos 2 <f> a 2 5 2 sin 2 6 involving circular functions if this quantity is positive, and loga- rithms if this quantity is negative. It is this double form which renders the process troublesome, if we adopt this order of integra- tion ; and D'Alembert discusses the matter at great length. The best mode would be to integrate with respect to <f> first ; this would lead to a result which we shall presently obtain in another way : but D'Alembert does not adopt this order of inte- gration. 647. Let us now consider the problem by the method III. of Art. 626. Suppose that instead of an ellipsoid we had an ellipsoid of revolution, in which the semiaxes are c, 6, and b. Then, by 414 D'ALEMBERT. Art. 255, the attraction on a particle at the end of the semi axis c would be 2 f 27r f2 sin# cos 5 h h & 2 + (c 2 - dud6 Then for the case of an ellipsoid, not of revolution, we must put p* instead of 6 2 , where t>* so that our formula becomes . ir r sin 6 cos 2 6 du dO 'o *' If we integrate with respect to 6 first, we shall have two forms, c a according as a is greater or less than unity ; and D' Alembert discusses the matter at great length. 648. But suppose we integrate the formula of the preceding Article with respect to u first. We have 2c sin 6 cos 2 2c sin 6 cos* 6 -, 8 2c sin ^ cos 2 6 ^ _ i\ s i n * cos 2 + -, cos' + T= 2c sin ^ cos 2 6 cos a 6 + -(sin 2 u + cos 2 ) sin 2 ^ u Q/ _ 2c sin ^ cos 2 6 (cos 2 + t sin 2 0) cos 2 u + (cos 2 + ^ sin 2 0) sin 8 M D'ALEMBERT. 415 Integrate with respect to u between the limits and and multiply the result by 4. Then we find that the required at- traction , ff _ sin 6 cos 2 dQ _ J o VO 2 cos 2 6 + c 2 sin 2 0) V(&" cos 2 + c 2 sin 2 0) Thus the attraction is made to depend on a single definite integral. We may say that this result is the point at which modern investigations have finally arrived. We shall presently see that D'Alembert absolutely rejected this important formula which was within his reach. 649. D'Alembert himself draws attention to the fact, that when we have to find the value of a double integral, the facility of the process may depend very much on the order in which we effect the two integrations. See his page 158. He makes this remark after he has considered a way of finding the volume of a right cone, by cutting it into hyperbolic slices, by planes parallel to the axis : this way is difficult, though the final result is necessarily very simple. 650. The third part of the memoir occupies pages 159... 207; it considers different ways of calculating the attraction of elliptic spheroids, and treats also of the attraction of some other spheroids. I will notice some of the more important points. We know that a plane may be so moved, parallel to itself, that all the sections which it makes of an ellipsoid shall be cir- cular sections. D'Alembert suggests the problem of finding the attraction of an ellipsoid at the" extremity of the diameter which passes through the centres of a series of circular sections. But the integrations are too complex to be worked out. See his pages 159. ..164. Let any segment of an ellipse revolve round its bounding chord ; then the attraction exerted by the solid thus generated on a particle at the extremity of the chord can always be found, 416 D'ALEMBERT. or at least expressed as a single definite integral without radicals. See D'Alembert's page 164. r/3 In fact, this attraction = STT r sin cos 6 d0, where ft is the Jo angle between the chord and the tangent to the ellipse at the h cos + k sin 6 origin ; and T = ; 2 /> . 7 : 3 ^ 2~/i a sin 2 6 + b sin 6 cos + c cos 2 A theorem which presents itself incidentally may be noticed : see D'Alembert's page 167. Take any diameter of an ellipse, and let a solid be generated by the revolution of one of the halves of the ellipse about this diameter : then the volume generated varies inversely as this diameter. If this diameter be called 21, and the axes of the ellipse be 2a and 25, the volume of the solid is ^ . of D'Alembert invites mathematicians to continue their attempts to express the attraction of an ellipsoid without the use of arcs of conic sections ; he says that the attempt does not appear to him hopeless : see his page 171. We now know that he was seeking what it is impossible to obtain. Plana has drawn attention to this passage inCrelle's Journal fur. ..Mathematik, Vol. xx. page 190. D'Alembert gives some simple examples of the process for the change of the independent variables in a double integral which Lagrange had developed in the volume of the Berlin Mtfmoires for 1773. See D'Alembert's pages 176 and 177. 651. We JQOW arrive at a very singular passage. D'Alembert in effect gives the process of our Art. 648 and rejects it as inad- missible. See his pages 177... 180*. His y is our 6. He says : J'avois imaging d'integrer d'abord la formule de la page 183 du Tome vi. de nos Opuscules en faisant varier u, et ensuite y, et j'avois cm trouver un resultat qui me conduisoit a une formule algebrique d'attraction pour les spheroi'des elliptiques. Comme cette methode pourroit en tromper d'autres, il ne sera peut-etre pas inutile de la de"tailler ici. D'ALEMBERT. 417 In his process there is nothing wrong in principle, but he has omitted a bracket in the third line of his Art. 147 ; thus his result is slightly inaccurate, He gives some invalid arguments against the method. Thus D'Alembert deliberately rejects one of the most important formulae of the subject, which in fact quite super- sedes a large part of the present memoir. This is perhaps the strangest of all his strange mistakes. 652. D'Alembert shews in his page 199 that a theorem given by Laplace in the Paris Memoir es for 1775 might be investigated with ease. Laplace himself found afterwards a much simpler demonstration than that which he originally gave : for this see the Paris Mtmoires for 1776, page 261. D'Alembert says in his page 221, with respect to Laplace and his two proofs of his theorem: Ce meme Academician, & qui j'avois communique ma demonstration tres-simple de son theoreme, en a aussi trouve* depuis une autre plus simple que la premiere, efc qu'il a lue a 1' Academic au inois de Juillet 1778. II m'a appris en meme-temps que M. de la Grange avoit aussi trouve de son cote une demonstration de ce theoreme general. The following is the theorem. Let the radius vector of a spheroid be 1 + a//,, where a is very small and //, a function of the colatitude ty. At any point of the surface let A denote the attraction along the radius vector, and B the attraction at right angles to the radius vector in the meridian plane from the pole : then will . d^ 2 3 d^fr" We shall demonstrate the theorem when we treat on Laplace's contributions to our subject. Suppose T the whole force along the tangent, <o the angular velocity : then to the order of approximation we regard T - B- A + a,' sin ^ cos f : d^jr in the small term we may put <r for A ; thus o T=-t?a^y +*> 2 sin^cos^ .......... (2). o d~yr T. M. A. 27 418 D'ALEMBERT. From (1) and (2) we have dA T ft) 2 . Let P denote the gravity, so that P=A - &) 2 sin' 2 ^ ; then (3) becomes 2 sin 2 ^) T a) 2 . - =--smfc thatls D'Alembert contemplates the theorem under the form (4), and puts it into words : see his pages 200 and 201. If the body is fluid and in relative equilibrium the condition T must be satisfied ; and thus the theorem is simplified. 653. On pages 391. ..392 of the volume D'Alembert suggests a process for the calculation of the attraction of a hemispherical mountain on a pendulum occupying any position close to the mountain ; but it is not fully intelligible, and nothing is really effected. He makes the erroneous statement that the direction at right angles to any radius of the mountain will also be at right angles to the radius of the Earth. 654. The eighth volume of D'Alembert's Opuscules Maihe'- matiques was published in 1780. A memoir entitled Nouvelles reflexions sur les loix de Ffyuilibre des fluides occupies pages 1...35 ; and there are some remarks relating to the memoir on pages 354. . 357. 655. This memoir is not very closely connected with our subject ; we will briefly indicate the nature of the topics discussed. We may observe that the old and obscure language with respect to fluid equilibrium is still retained ; no advantage is taken of the capital improvement effected by Euler in introducing the notion of pressure and its appropriate symbol p. D'Alembert notices an objection which he says an able mathe- matician had brought to him. It amounts to this. Suppose for D'ALEMBERT. 419 simplicity the density of the fluid to be uniform ; then we have shewn in Art. 394, that dY dX m dx ~ dy ' but in the demonstration small quantities of the second order have been neglected : thus we may be in doubt whether any inference from this result is rigorously true. D'Alembert's words adapted to the notation and diagram of Art. 394 are: ...il est certain que Xdx ne repr6sente la force du canal PS qu'a un intiniment petit du second ordre pres, puisqu'on neglige les quantites infmiment petites du premier ordre qui entrent dans X, pour en expritner la valeur le long du canal PS ; il est certain aussi qu'il en est de meme de Ydy \ ne peut-on pas coiiclure dela, m'a objecte J V" /7V" un habile Mathematicien, que 1' equation - - -=- ne repr6sente l'e"qui- libre du canal rectangulaire PQRS, qu'a un infiniment petit du second ordre pres, et qu'ainsi elle ne represente pas rigoureusement Tequilibre, qui doit exister rigoureusement entre les parties du fluide, et qui seroit necessairement trouble, s'il n'etoit pas tel? D'Alembert discusses the matter in his pages 2... 8. l)'Alembert considers whether it is necessary that X and Y should be continuous ; that is whether throughout the fluid X should always be the same function of x and #, and also Y the same function of x and y. He maintains correctly that this is not necessary : see his pages 9... 15. But he seems on his page 355 to lose faith in his own demon- stration. In his pages 16... 20 he adverts to a supposition he had formerly made with the view of giving greater generality to the equations of fluid equilibrium : see Art. 397. In effect he now abandons that supposition. In his pages 20... 26, to use modern language, he makes some remarks on the equations of fluid equilibrium, when referred to polar coordinates ; he had formerly considered this topic: see Art. 574. His pages 26... 28 he devotes to shewing that if a fluid occupies an infinite tube, and a finite portion of the fluid be put in motion, 272 420 D'ALEMBERT. no sensible movement in the mass will be produced. It does not seem to me that the investigation is of any value. In his pages 28... 30 he professes to demonstrate the statement, commonly admitted by writers on hydrostatics, that if a fluid mass be in equilibrium any portion of it may be supposed to become solid without disturbing the equilibrium. The demonstration does not seem to me of any value. We have three last remarks in conclusion. On page 30 he says : Terminons ces recherches par quelques reflexions sur la loi de la compression de Fair en raison des poids dont il est charge. Then on page 32 he says : Nous ajouterons ici en finissant, une remarque a laquelle il est bon de faire attention dans la gradua- tion des barometres. And on page 33 he says : Je terminerai ces recherches par une nouvelle remarque sur la the'orie de Nquilibre des fluides. This new remark however is substantially old, having been given in page 206 of the TMorie de la Resistance des Fluides: see Art. 448. 656. In pages 292... 297 of the eighth volume of the Opus- cules Mathematiques we have a memoir entitled Sur la Figure de la Terre ; and some remarks on it are given on pages 389... 392: these form D'Alembert's last contribution to our subject. 657. We have already observed that D'Alembert having arrived at a certain equation shewed that it would sometimes have two roots ; but left for others to demonstrate the proposition that there could not be more than two roots ; and this was first established by Laplace : see Arts. 581 arid 585. D'Alembert says : M. de la Place m'en a communique une demonstration assez simple qui m'en a fait aussi trouver une trs-simple, presque sans aucun calcul. D'Alembert's demonstration is ingenious in principle but unsound. In equation (1) of Art. 581 put x for X ; thus we get D'ALEMBERT. 421 Let y be an ordinate corresponding to the abscissa #, and let the curve be drawn whose equation is Again let 77 be an ordinate corresponding to the abscissa x, and let the curve be drawn whose equation is 77 = tan" 1 x. We have to shew that the curves cannot intersect more than twice for positive values of x, besides the intersection at the origin. We have y and thus when x is very small x 3 Also when x is very small Thus when x is very small y is greater than 77 ; and so near the origin the first curve is above the second. When x is infinite y is infinite and 77 is finite. Thus if the curves intersect at one point, say x v they must intersect at another say x v At this second point therefore -jf- will be greater than -^ . To ensure that the curves never intersect again we have only to shew that ~ -^ is always positive when x is greater than # 2 ; for if this be the case, y is always greater than rj if x is greater than # 2 . Now D'Alembert says that '- -y- is of the form dx dx and this is true. Then he asserts that every quantity of the form Ax 9 + BX* + Cx* + D, which is positive for a certain value of x, will be positive if x is increased. His words are : 422 D'ALEMBERT. . . .toute quantite de cette forme Ak 6 + Bk* + Ck a + D, qui sera positive pour une certaine valeur de k, doit 1'etre si on augmente k] car cette quantite est to uj ours = Ak 2 (k* + E)* + G, qui augmente quand k augmente. But this statement is untrue. In the present case -f- -^ OjX GLOC is positive when x is very small, but it is not always positive : it must be negative when x = a? r The demonstration may be made sound by shewing that in the present case the values of A, B, C, D are such that Asi+Btf+Csf+D is always positive when x is greater than # 2 . This method is really adopted by Cousin in his Astronomic Physique, page 148. But we do not require the values of A, B, C, D to establish the point ; it is sufficient to observe that D is zero : this is obvious from the fact that when x is very small we must have dy drj _ d 2<p 8 dx dx dx 9 Since then D 0, we have dxdx~ (1~+ oj 2 ) (9 now we know that the quadratic expression Ax* + Bx z + C cannot change sign more than twice ; and in the present case the sign is positive when o? = 0, negative when x = x lt and positive when x x, 2 ; therefore the sign must always be positive when x is greater than x z . 658. The memoirs of D'Alembert on our subject which we have thus analysed in our Chapters XIII. and XVI. occupy about 700 pages in their original form. But the amount of im- portant matter which they contain is not in proportion to their great extent. Probably the researches in the third volume of the Recherches...Systeme du Monde are the most valuable. The sixth volume of the Opuscules MatMmatiques contains much interesting matter ; but this matter is rather of a speculative kind than of physical importance. D'ALEMBERT. 423 On the whole we may sum up D'Alembert's contributions to our subject thus : He shewed how to calculate the attractions of a nearly spherical body of a form more general than an ellipsoid of revolution : see Art. 452. He drew explicit attention to the fact that more than one oblatum would correspond to a given angular velocity, a fact which had indeed been implicitly noticed before : see Art. 580. He considered the action of a distant body or bodies on a mass of rotating fluid supposed in relative equilibrium : see Arts. 596... 630. On the other hand we must observe that there are numerous and striking faults. Laplace, referring more especially to the Eecherches. . . Systeme du Monde, says : Les recherches de D'Alembert, quoique gdnerales, manquent de la clarte' si ne*cessaire dans les calculs complique's. Hecanique Celeste, Vol v. page 8. The full import of the criticism becomes apparent when we remember that with French writers clarte is the supreme indispensable requisite : want of clearness with them is on the same level as want of utility with Englishmen, or want of learning with Germans. The errors of D'Alembert are certainly surprising ; they seem to me to indi- cate that he was little in the habit of enlarging his own views by comparing them with those of others. His criticisms of Clairaut prove that he had not really mastered the greatest work which had been written on the subject he was constantly studying. His readiness to publish unsound demonstrations and absolute errors is abundantly shewn in the course of our criticism : see for instance Arts. 576, 651, and 657. On the whole the blunders revealed in the History of the Mathematical Theory of Probability, and in the present History, constitute an extraordinary shade on a fame so bright as that of D'Alembert. CHAPTER XVII. FRISI. 659. IN the present Chapter I shall give some account of three works by Paul Frisi. As I have stated in Art. 532, I have not seen the first publication by Frisi on our subject; but proba- bly it was incorporated in his later works. 660. The first of these works is entitled De Gravitate Libri Tres. This was published at Milan in 1768: it is a quarto volume of 420 pages, besides 12 pages which contain the Title, Dedication, Preface, and Index; there are six plates of figures. This work forms a treatise intended for didactic purposes, the object being to conduct a student with elementary mathematical knowledge through a course of Mechanics and Physical Astro- nomy : see the first page of the Preface. The two volumes published by Frisi about six years later, under the title of Cosmo- graphia, may be regarded as an improved and enlarged edition of the present work. The part of the volume with which we are concerned consists of pages 135... 189; they form the first four Chapters of Frisi's Second Book. The pages 135... 145 are introductory. They contain an out- line of the facts then known as to the lengths of degrees and to the lengths of the seconds pendulum. 661. The first Chapter is entitled De Figura Terrce, it occu- pies pages 146... 154. This Chapter contains approximate formulae for the lengths of a degree of meridian and of a degree of longitude. From these FRISI. 425 formulae, and the results furnished by observation, the ellipticity of the Earth may be deduced. Frisi maintains that all the measurements hitherto made agree reasonably well with the ratio of the axes assigned by theory for an oblatum of fluid, "namely that of 231 to 230. Some erroneous statements occur in the second Corollary on page 151. Frisi has given a formula for determining the ellip- ticity from the lengths of a degree of meridian in two different latitudes. Then he says that if the arc in Lapland and the arc at the Cape of Good Hope be taken the ellipticity deduced is but in the Cosmographia, Vol. II., page 97, he gives the correct result, namely about - . The error probably arose from taking ' the ellipticity which Boscovich had deduced from the arcs in France and South Africa by mistake, instead of that which was deduced from the arcs in Lapland and South Africa: see the supplement by Boscovich to Stay's poem, Vol. II., page 408. Again Frisi gives TTTT as the ellipticity deduced from the arcs in Peru and South Africa; but in the Cosmographia, Vol. IL, page 96, he gives the correct result, namely about =-. The error probably arose from simply copying one made by Boscovich : see Art. 508. Frisi's first Chapter closes thus: Quam ex amplissima Pensilvanise planitie Clariss. Mason, et Dixon afferent mensuram gradus figuras terrestris inquisition t novam lucem affimdent. Interim certum manet Terram sub sequatore, et polari circulo, et in meridionali Africse, et Gallise Narbonensis parte, atque in Anglia etiam, et Stiria, ac Pedemonte, a figurse sphseroidicae, et propor- tionis assumptse hypothesi non magis recedere, quam ut in minimos errores observationum differentia omnis refundi possit. The anticipation as to the light to be derived from the American arc, has scarcely been realised ; for this arc has not been received with much confidence: see Bowditch's translation of the Mecanique Celeste, Vol. n., page 444. 426 FRISL 662. The second Chapter is entitled De cequilibrio particu- larum sese trahentium; it occupies pages 154... 164. This Chapter contains a demonstration of the proposition that an oblatum "is a form of relative equilibrium for rotating fluid; the method is that of Maclaurin and Clairaut: see Art. 318. We have, in this Chapter, some extensions to ellipsoids in general of results which had already been established for ellipsoids of revolution : seethe Corollary ir. on page 157, and the Corollary II. on page 158. Thus Newton had shewn that a shell bounded by homothetical ellipsoids of revolution exerts no attraction on a particle placed within the inner surface. Frisi shews that this is true for a shell bounded by homothetical ellipsoids when the particle is on the inner surface. He does not expressly shew that this is true when the particle is within the inner surface ; but it was quite in his power to infer this from what he had already given. This seems to be the first introduction of the ellipsoid, as dis- tinguished from the oblatum and the oblongum, into our subject. D'Alembert afterwards considered the matter in the sixth volume of his Opuscules Mathematiques : see Art. 615. Frisi also alludes to the results which will follow when the fluid oblatum is disturbed by the action of one or two distant attracting bodies, like the sun and the moon. His process how- ever is brief and not very satisfactory. This matter was after- wards discussed in detail by D'Alembert in the sixth volume of his Opuscules Mathe'matiques : see Chapter xvi. At the end of the Chapter Frisi refers to Maclaurin, Simpson, Clairaut, and Newton. The last is styled vir longe omnium inge- niosissimus: these words are omitted in the corresponding passage of the Cosmographia. But in both works Frisi says: ...ut recte propterea dixerit Daniel Bernoullius, 8.' cap. 2. de fluxu, et refluxu maris, Newtonum trans velum etiam vidisse, quse vix ab aliis microscopii subsidio discerni possunt. We have given the original words of D. Bernoulli in Art. 501. 663. The third Chapter is entitled De sphcerarum, splweroi- dwnque attractione; it occupies pages 164... 175: but the pages FIUSI. 427 170. ..173, which are rather difficult, do not belong to our subject, and are removed to a more appropriate place in the Cosmographia. Here we have an exact investigation of the attraction of a spherical shell on an internal particle ; and an application to the case in which the particle is on the surface of the shell, or forms a component of the shell. The process, like others in the Chapter, really involves the Integral Calculus, though without its notation. Next we have an approximate investigation of the attraction of an oblatum on a particle situated on the prolongation of the axis of -revolution | the result is correct to the first power of the elliptic! ty. Then we have an approximate investigation of the attraction of an oblatum on any external particle; this problem is treated in Clairaut's manner: see Clairaut's pages 236... 239, or Art. 335. Frisi refers to the criticism of Short and Murdock on his sup- posed discovery of an error in Newton : see Arts. 533 and 534. Frisi however does not admit the accuracy of the criticism ; he says: Nsevum hujusmodi cap. 6. dissertations de Figura Terrse a nobis jam adnotatum, in Transact, anni 1753. excusare voluerunt Clariss. Short, et Murdock, postrema Newtoni verba in eadem ratione quam proxime intelligentes de rationis continuitate, non de identitate cum ratione semiaxium : qui tamen sensus allati textus minime videtur esse. 664. The fourth Chapter is entitled De cequilibrio, et lege ter- restrium ponderum ; it occupies pages 175... 189. Here we have first a proposition and corollaries which belong rather to a rude theory of the tides than to our subject. Next we have an approximate investigation of the ratio of the axes, in order that an oblatum of rotating fluid may be in relative equilibrium. Then it is shewn that if the Earth be homogeneous, or be composed of spheroidal strata, the weight of a given body on the surface of the Earth will increase in passing from the equator to the pole; the increment varying as the square of the sine of the latitude. 428 FRISI. For a particular case Frisi finds that we may reasonably satisfy the observations by supposing the Earth to consist of a sphere having the minor axis for diameter, and of an outer portion ; each of the two portions being homogeneous, but the density of the sphere to the density of the outer portion as 1 -f -J is to 1. See his pages 183... 185. On his page 186, Frisi draws attention to a point as to which he differed with Newton. He had already referred to this by anticipa- tion on his page 174, where he says Omnino falsum est illud, quod in Prop. 38. Lib. 3. assumpserat Newtonus,... We will. return to this point when we give an account of Frisi's Cosmographia. 665. On his pages 224... 235 Frisi has a Chapter entitled De variationibus Maris, quce oriri possunt ex Sole aut Luna. The first half of this Chapter bears rather more on our subject than the title might seem to indicate; but we will reserve our notice of it until we speak of the Cosmographia. 666. Frisi himself gave an account of the contents of his work before it was published; this account is contained on pages 514. . .530 of the Bologna Commentarii, Vol. V., part 2, 1767. This account adds nothing to our subject. Frisi, on page 522, draws attention to the two points at which he differs with Newton : see Arts. 663 and 664. 667. Judging from the part of Frisi's work which I have thus had to examine, I should say that it may be considered to have formed a reasonably good elementary treatise at the time of its appearance. It contains however none of the higher researches which Clairaut had given as to the Figure of the Earth, when supposed to be heterogeneous ; and thus the promise held out in the Preface of conducting the reader to the summit of physical astronomy ad summum Physicse celestis apicem is scarcely ful- filled. ' 668. We have now to notice the second work by Paul Frisi. It is in two quarto volumes. The first volume is entitled Cosmo- graphics Physicce, et Matfiematicce Pars prior Motuum periodicorum theoriam continens. The second volume is entitled Cosmographies FRISI. 429 Physical, et Mathematics Pars altera De Rotationis Motu et Phce- nomenis inde pendentibus. The work was published at Milan; the first volume is not dated ; the second is dated 1775. The first volume contains 266 pages, besides a page of errata, and the Title, Dedication, and Index on 6 pages. The second volume contains 276 pages, besides the Title, Dedication, and Index on 6 pages. Each volume has three plates of figures. 669. It is well known that in what is called the Jesuits' edition of Newton's Principia, there is a note by the editors in which they profess their submission to the decrees issued by the supreme Pontiffs against the motion of the Earth, although in commenting upon Newton they were obliged to adopt the same hypothesis as he did. I do not know at what date these decrees of the supreme Pontiffs were first allowed to be disregarded. Certainly in the present work Frisi has no hesitation in adopting the truth as to the Earth's motion; his language seems much more decisive than it was in his former work. We have the following words on page 28 of Vol. I. of the Cosmographia : Galilseus Mar tern, et Yenerem moveri circa Solem certissime ex eorumdem phasibus collegit. Totum vero Telluris motse sistema novo hoc analogic argumento confirmatum ita in dialogis vindicavit, adorna- vitque, ut, qua in physicis rebus certitudine fieri poterat, ostenderit Planetas quinque primarios simul cum Terra motu periodico ab occi- dente in orientem revolvi circa Solem in planis transeuntibus per Solera ipsum, et parum dehiscentibus a se invicem : Lunam ab occidente pariter in orientem revolvi circa Solern,... The context shews that the last word Solem is a mistake for Terram. 670. The Cosmographia may be considered as an enlarged edition of the treatise De Gravitate, of which we have already given an account. The part of the Cosmographia with which we are concerned consists of pages 83... 142, and 207... 21 9 of the second volume. 671. The pages 83... 142, form the Second Book of the second volume. This Book consists of an introductory portion followed by four Chapters. 430 FRISI. The introductory portion occupies pages 83... 92. This gives an account of the various measurements of arcs on the Earth's surface up to the current date. The following important passage relative to the errors which might arise from the use of a zenith sector, occurs on page 88 : ...certo autem ostendit Clariss. Maskelinius cum in expeditions acl insulam S. Helenae pro parallax! Sirii, aliisque Caillii observationibus recognoscendis suscepta deprehendit iis sectoribus, quibus Maupertuisius, aliique, ad mensuram graduum usi fuerant, frictionem fili ex instrumenti centre suspensi errorem 3", aut 4" in partes adversas quandoque pa- rere : ut fuse a summo ipso Astronomo cum Grenovicii essem accepi. The substance of this passage occurs also in the De Gravitate, page 140; but the words from ut fuse to the end are not given there. Frisi notices that the arc measured at the Cape of Good Hope by La Caille, was longer than might have been expected from the results of other measurements. He suggests that this may be owing to the circumstance that the continent of Africa supplies an excess of matter toward the end of the arc which is nearer to the equator when contrasted with the ocean near which the other end of the arc is situated. Thus the plumb line at the end of the arc which is nearer to the equator may be considered to be affected as it would be by a range of mountains at the equator; so that the amplitude of the arc would be rendered a little too short. This suggestion was also made by Cavendish ; see the Philosophical Transactions for 1775, page 328. We have spoken of the modern remeasurement of the South African arc in Art. 542. 672. The first Chapter is entitled De dimensione graduum, et, quce inde colligitur, Figura Terrce ; it occupies pages 9 2... 104. This Chapter corresponds to the first Chapter which we ex- amined in the former work ; see Art. 661. Frisi maintains, as before, that all the measurements hitherto made agree reasonably well with the ratio of the axes assigned by theory for an oblatum of fluid, namely that of 231 to 230. FRISI. 431 673. The second Chapter is entitled De cequilibrio particula- rum omnium sese trahentium: it occupies pages 104... 11 3. This Chapter corresponds to the second Chapter of the former work: see Art. 662. There is a slight mistake in the second corollary on page 112. The mistake is not of much importance, but the correct expres- sions involved are often useful: see Art. 596. Suppose a body acted on by a very distant particle of mass M. Take a fixed point in the first body as origin; and let the axis of x pass through the second body. Let k represent the distance of the particle of mass M from the origin. Then the action of M at a point (a?, y) will be equivalent to M(k x] -ft parallel to the axis of x, and ^- parallel to the axis of y\ where R 2 = (Jc x)' 2 +y* : both resolved attractions being estimated outwards. O* 97 Expand and neglect powers of y and | above the first. Thus we obtain ^ + -^- parallel to the axis of x, and ~ parallel to the axis of y. The force p * s constant. Thus in the language of the Planet- ary Theory we may say that we have as disturbing forces, -^ K parallel to the axis of x, and ~ parallel to the axis of y. Frisi's mistake consists in changing the coefficient 2 to 3. We may if we please arrange these disturbing forces differently. 2Mx 3Mx MX 3Mx bmce -p- -, 3 --- TJ- , we may say that we have -p- parallel to the axis of #, besides - and -p parallel to the axes of x and y respectively : then the latter two may be combined into a single force towards the origin. Thus finally we have , 3 X parallel /v to the axis of x, and ^f, + y ^ towards the erigin. 432 FRISI. Or we may if we please arrange these disturbing forces in . My another way. bmce - -jf = ^ -- j~- , we may say that we have ~ parallel to the axis of y, besides ~- and parallel to the axes of x and y respectively : then the latter two may be combined into a single force towards the origin. Thus finally we have - -f{? parallel to the axis of y, and 2 ^Cf + .?') fa k from the origin. 674. The third Chapter is entitled De sphcerarum, spheroi- dumque attract ione ; it occupies pages 114... 123. This Chapter corresponds to the third Chapter of the former work : see Art. 663. Frisi retains his opinion noticed in that Article as to the supposed error of Newton. 675. The fourth Chapter is entitled De Planetarwn Figura, quce ex cequilibrii legibus colligitur; it occupies pages 123... 142. This Chapter corresponds to the fourth Chapter of the former work : see Art. 664. The - of Art. 664 is now replaced by -^ . 676. I will now explain the difference between Frisi and Newton to which I have alluded in Art. 663 : Frisi refers to it on his page 135. Let m denote the mass of the moon, M that of the Earth, k the distance between their centres. Suppose the moon to be a homogeneous fluid ; then the surface of the moon may be in the form of an oblongum with the longer axis directed to the Earth. Let b denote the minor semiaxis of this oblongum, and b+h the major. Suppose the centre of the moon brought to rest ; then we may consider that besides the attraction of the Earth, there is a central disturbing force, and also the disturbing force 8 as in Art. 673. Then if we proceed^as Newton did for determining the figure of FRISI. 433 the Earth, or in some more analytical method, we shall obtain approximately h _ 5 fr 3 _ 15 M b* = 4" m = ~4'm'k*' sothnt There is no difference of opinion as to this result ; it agrees with one obtained by D. Bernoulli in his Essay on the Tides, Chapter iv. Article 8. It is also consistent with Newton's result in the Principia, Book ill. Proposition 36. The simplest way of connecting the result with Newton's in- vestigations is to adopt the last method of arranging the disturb- ing forces given in Art. G73 ; so that we have a central force and also a force ' parallel to the axis of y. Then in Art. 28 cor- responding to a centrifugal force which may be denoted by o> 2 y we obtained an oblatum in which e = -j ; and now corresponding to a disturbing force -- J we obtain an dblongum with a similar value for the ellipticity. Now proceed in a similar way to determine the figure of the Earth, supposed fluid, under the action of the moon. Let B and B-+H denote respectively the semiaxes of the ob- longum ; then we have Hence, by division, H'~\m> This result is given by Frisi. But Newton in his Principia, Book ill. Proposition 38, asserts in fact that k M b H m'~B' T. M. A. 28 434 FRISI. It is clear that Frisi is right ; but I do not know of any com- mentator on Newton, who has accepted the correction. Some further information on the subject will.be found in a paper published in the Monthly Notices gf the Royal Astronomical Society, Vol. XXXIL pages 234... 236. 677. In his pages 140.,. 142 Frisi treats on the attraction of mountains; he refers to what D'Alembert had given on the subject in the sixth volume of his Opuscules Mathematiques. Frisi in these pages uses the notation of the Differential Calculus which does not occur in the other parts of his work that have come under our examination. Frisi throws doubts on the genuineness of those observations to which D'Alembert drew attention : see Art. 594. 678. Frisi's pages 207... 219 form a Chapter entitled De cequilibrio fluidorum nucleos sphceroidicos circumambientium. This Chapter presents in an improved and enlarged form the propositions bearing on our subject to which we referred in Art. 665. Frisi first shews that the attraction at any point of the surface of a nearly spherical oblatum or oblongum consists of a central force together with a small force parallel to the major axis. Then he considers the case of a spherical nucleus surrounded by fluid, the whole rotating with uniform angular velocity. Next he sup- poses the nucleus to be an oblatum or an oblongum ; this investi- gation includes the preceding as a particular case. The result at which Frisi arrives amounts to the same as that which I have stated at the end of Art. 374 ; his investigations are fairly satis- factory. On his pages 215 and 216, Frisi arrives at results respecting what we should now call the stability of the relative equilibrium, which resemble those of D'Alembert : see Art. 567. Frisi's in- vestigations on this matter however, as might be expected, are rather rude ; they were not given in his former work, and were doubtless suggested by the sixth volume of D'Alembert's Opuscules Mathematiques, to which Frisi refers on his page 219. Frisi refers also on this page to Boscovich, respecting the same matter, saying FKISI. 435 " de quibus casibus plura ingeniose scripserat Boscovichius." Frisi however does not say that his own conclusion does not agree with that of Boscovich ; the latter, as we have stated in Art. 470, held that the oblongum could not be a stable form, whereas Frisi holds with D'Alembert that it is so in a certain case : see Art. 590. On his page 210 Frisi attempts to investigate the approximate expressions for the attraction on a particle outside an oblatum resolved at right angles to the straight line drawn to the centre, supposing the particle very near the plane of the equator : but I cannot consider his process satisfactory: see Art. 321. 679. Frisi's Fifth Book, entitled De Atmosphcera Planetarum, occupies pages 231... 271 of the second volume of his Cosmographia. It is not really a part of our subject, and I have not examined it throughout. I will however make some remarks on certain passages. Frisi deduces in a satisfactory manner the result which we may thus express in modern language: if we omit all consideration of centrifugal force the height of the atmosphere will be infinite. But then he seems to be frightened at his own result, and makes a remark which amounts to saying that his investigation does not hold. See his page 240. The subject seems to have been regarded as paradoxical by some of the mathematicians of the eighteenth century. See for instance Fontana's Ricerche sopra diversi punti..., pages 89... 105. Fontana refers to an error committed by David Gregory; Frisi without mentioning a name seems to refer to the same point in his second Corollary on page 239. Fontana also finds fault with section 36 of a memoir by Play- fair, in the Edinburgh Transactions for 1788; but it seems to me that Fontana misunderstands what is said : Playfair wishing to shew that the height of the atmosphere. is infinite attains his end by supposing that the density is zero, for then his formula gives the inadmissible consequence that the radius of the earth is infinite. He is however not so clear as he might be on the matter, and Fontana takes him to make the hypothesis that the radius of the earth is infinite. Fontana's words are : 282 436 ERISI. II Sig. Giovanni Play fair... inferisce, che il semidiametro terrestre r e infinito, e di qui che the athmosphere on this supposition admits of no limit, illazione visibilmente assurda essendo contro il fatto, e la iiatura delle cose il dare al semidiametro della terra una lunghezza infinita. Frisi obtains, in his pages 254 and 255, an equation to deter- mine the limit of the atmosphere; the equation representing the generating curve of the bounding figure. This equation is where a represents the equatorial semiaxis. This agrees with what we have already given : see Art. 640. Frisi, in fact, deter- mines one of the two constants there occurring, inasmuch as he supposes the surface to pass through the points where the centri- fugal force becomes equal to the attraction. Frisi says in his page 257, with respect to this equation, ...atmospherse figuram ex limitibus altitudinis sic deduximus ut rami omnes excurreiites ad infinitum, et casus alii ramorum duplicium prse- cluderentur* quos D. Mairan in tractatu de Aurora Boreali enunie- raverat. 680. A collection of works by Frisi was published at Milan in three volumes quarto; the first volume appeared in 1782, the second in 1783, the third was issued by the author's brothers in 1785 after his death. We are concerned only with the third volume of these works, which is entitled Paulli Frisii Operum Tomus tertius Cosmogra- phiam Physicam, et MatJiematicam continens. It contains 561 pages, besides the Index on 3 pages, and the Title and Dedication on 6 pages ; there are 3 plates of figures. 681. The Second Book of this volume is entitled De Figura Terrce et Planetarum; it occupies pages 11 7... 184. This may be described as a republication of the Second Book of the second volume of the Cosmographia, with some omissions and some additions ; the changes however are of little importance. We may therefore refer to the notice already given of the Cosmographia in Arts. 668... 679; and shall only add a few remarks on some points of interest. FRISI. 437 682. On his pages 164 and 165, Frisi discusses the problem of the solid of revolution of given volume and maximum attraction. He arrives at the following differential equation for the generating (y 2#\ s^- 5- j dx, which is correct. But three observations 6x 6y/ suggest themselves. (1) Frisi makes no reference to an incorrect investigation which he had formerly given; to this we shall return in the next Chapter : see Art. 686. (2) In enunciating the problem he implies that the gene- rating curve is to pass through two fixed points; but he pays no attention to this condition in his solution. If I had been acquaint- ed with this passage of Frisi's work when I published my Re- searches in the Calculus of Variations I should have noticed it then. Compare page 123 of that work. (3) Frisi in a Corollary adverts to the solution given by Silvabelle: see Art. 531. Frisi seems to think that the results obtained by himself and Silvabelle do not agree: for he uses the words "formularum diversitatem." But Silvabelle's result is a*x= (ic 2 + ?/ 2 )-, where a is a constant; and if we differentiate this, and eliminate a, we obtain Frisi's result. Frisi then objects to Silvabelle's solution. The objection amounts to saying in modern language that Silvabelle confounds dy with Sy. Silvabelle's process however is quite sound, if we are careful to understand it properly. 683. We may notice the points in which Frisi makes special reference to Newton. (1) The passage which I have quoted in Art. 662 now ap- pears on page 153, thus : Daniel Bernoullius... acute dixerat Newtono trans velum etiain ap- paruisse quae vix ab aliis microscopii subsidio distingui possunt. (2) Frisi retains his opinion as to the supposed error of Newton: see Arts. 663 and 674. (3) Frisi, with more justice, retains his opinion as to the other point on which he differed with Newton: see Art. 676. Frisi 43S viiisi. savs, in reference to tins, on his page 170, a (jui Xewtoni locus a nemine antea fuerat emcndatus." (584. "We may observe that the names of La orange and *s ^ O Laplace art 1 now mentioned hy Frisi: see his pages lo.S and 1G(5. In Frisi's pa^es 100.,, 20}-, AVC have reproduced with some additions the matter contained in pages 207... -11") of the Cosmo- ia: see Art. (57S. All the three works hy Frisi -which we have noticed are printed on stout durable paper. Either the ^oneral public must have received them with a favour not usually bestowed on mathe- matical treatise's, or they must have obtained the private patronage of wealthy persons; for the expenses of producing them could scarcelv have been otherwise sustained. CHAPTER XVIII. MISCELLANEOUS INVESTIGATIONS BETWEEN THE YEAES 1761 AND 1780. 685. THE present Chapter will contain an account of various miscellaneous investigations between the years 1761 and 1780. The first three of Laplace's memoirs relating to our subject were published during this period, but it will be convenient to defer our notice of them until the next Chapter. 686. In the Novi Commentarii...$t Petersburg, Vol. VIL, which is dated 1761, there is a paper by Frisi, entitled De Pro- blematis quibusdam isoperimetricis : it occupies pages 227... 234 of the volume. This paper belongs to the early history of the Calculus of Variations, and not to our subject. I advert to it however be- cause on his last page, Frisi alludes to Silvabelle's problem and two others of the same kind, but without referring to Silvabelle : see Art. 531. In particular, Frisi states definitely his result for Silvabelle's problem. But this result is wrong ; and in fact the whole page is vitiated by an error which occurs at the top. It will be found that in his notation he has neglected to allow for the change of TL into PL. 687. An academical essay was published at Tubingen in 1764, entitled Dissertatio Physico-Mathematica de ratione pon- derum sub polo et cequatore Telluris...auctor Wolff g angus Ludovicus Krafft This consists of 28 pages in small quarto, with a page of diagrams. It appears from the last page of the essay that the father of the author had been a professor at Tubingen. 440 MISCELLANEOUS INVESTIGATIONS BETWEEN 17C1 AND 1780. 688. The mechanical principles involved in the essay are not always sound. Thus in the first paragraph the author seems to think that the time of Jupiter's rotation on his axis is determined by Kepler's Third Law ; for he says : Pan modo in Jove, et multo quidem major, diametrorum inae- qualitas a Cassini et Flamsteedio detecta est, civjus diurnam circa axem con version em cum plus, quam duplo celeriorem nostra, ostendat sagacissimi KEPPLERI regula, quadrata temporum periodicorum esse ut cubos distantiarum a sole... 689. For another example, we may take some remarks which the author makes involving centrifugal force. Suppose a sphere of radius a to rotate with uniform angular velocity; let /denote the attraction at any point of the surface, and </> the cen- trifugal force at the equator: then at a point on the surface in lati- tude X the gravity will be approximately / < cos 2 X. So far Krafft is correct. Now produce the radius vector of the place to a point at the distance x from the centre. Then he reduces the expres- sion for gravity in the ratio of a 2 to a?, and takes for the centrifugal force resolved along the radius vector the usual expression : thus he obtains for the gravity - 2 (/ <cos 2 X) <cos a X. But it is X CL obvious that he has thus introduced the centrifugal force twice, once erroneously and unnecessarily. The expression for gravity should be -3 <cos 2 X. This error occurs on pages 8 and 10. x a On page 10 he assigns the distance from the centre at which gravity would vanish; but the result depends on Ids erroneous formula, and is therefore wrong. 690. But it is curious that Krafft avoids this mistake in a problem which he discusses. The problem would be expressed thus in modern language : to find the lines of force outside a sphere, supposing that the sphere attracts, and that there is also a force of the nature of centrifugal force. Take the axis of x for that which would correspond to the axis of rotation ; let MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 441 Then we have to determine a curve from the equation dx dy ~Y V ' JL\- -* f(ydx /y * / ^ 1 bus vdx xdii that is f = cydx, - t* where c is put lor 7^. Krafft obtains this differential equation; to integrate it he assumes y = ^-~- , this gives 2 2 . 1 + hence ^ + constant = log - o J. ~~ that is -5- + constant = log 7-3 77-2 ^ . 3 ^VC^+rT-a V(^ +3/0 Krafft gives the second term on the right-hand side incorrectly : see his page 12. Krafft does not enunciate his problem in our modern language. According to him a tower is to be built on the Earth's surface, in such a manner that there is to be no force at any point tending to overturn it : in other words, the force at any point of what we may call the axis of the tower is to be along the tangent to the axis. 691. Krafft shews that the increment of gravity in proceeding from the equator to the pole varies as the square of the sine of the latitude very approximately; his method closely resembles that given by Boscovich, and is liable to the same remark : see Art. 467. Moreover, the theorem which Krafft uses as the foundation of his method is that to which we have drawn attention in Art. 34, and this assumes that the Earth is in the form of a fluid rotating in relative equilibrium. But Krafft has said nothing about fluidity, 442 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. so that an incautious reader might suppose him to be affirming some proposition relative to the attraction of an oblatum. 692. The main part of the essay however is the determination of the attraction of an oblatum at the pole and the equator ; this is finally accomplished to the order of the square of the ellipticity : see his pages 14... 20. He first calculates the attraction of an elliptic lamina on a particle directly over the centre of the lamina. Thus his problem is the same as Euler had already considered, and Krafft investi- gates it in a similar way, but there is no reference made to Euler : see Art. 229. 2 72 If we put X for 2 2 ^ , we shall find by that Article that if a \p ~f~ c j we neglect X 3 and higher powers of X, the attraction + c 2 ) Of the three integrals which this involves, Krafft works out the first in the laborious way which Euler adopted ; he merely states the values of the other two integrals. We have obtained the first and the second by a simple method in the Article cited. The third may be easily given. We have 2 dx = ( o? c 2 4 Jo c 2 + ^ Jo V Jo Jo = 16~~~4~ + T \ "T" " 1 J* Thus Krafft obtains a very approximate value of the attraction of the elliptic lamina ; but he speaks on his page 18 as if he had thus obtained the accurate value. Then knowing the attraction of the elliptic lamina, he proceeds to calculate that of an oblatum at the equator and at the pole ; his investigation is rather intricate, MISCELLANEOUS INVESTIGATIONS BETWEEN 17C1 AND 1780. 443 but it is correct, and his final results are in exact agreement with those given towards the end of Art. 229. It is strange however to see this tedious approximate solution of the problem of the attraction of an oblatum at its equator and pole so long after the exact formulae had been obtained by Maclaurin and by Simpson. 693. Krafft says that if we take the polar axis to be to the equatorial as 100 is to 101, we find that the attraction at the pole is to the attraction at the equator as 134396 is to 134129, that is approximately as 509 is to 508. It seems to me that by his formulae the ratio should be that of 352790 to 352091, that is approximately that of 505 to 504: moreover the approximation of 509 to 508 does not follow from his first figures. 694. Some miscellaneous observations on mathematical topics occupy pages 21... 2 7 of the essay. They do not seem to contain anything of interest except a statement relative to the series 1, 3, 4, 7, 11, 18,... in which each term is the sum of the two preceding. This series, according to Krafft, was introduced by Daniel Bernoulli in his Exercitationes Mathematics, who gave a formula for the sum of a finite number of terms, the value of the last term being given. Let p denote the last term, then the sum is - {3p 6 + V(% 2 20)}, where in the ambiguity the upper or the lower sign is to be taken according as the number of the terms is odd or even. Krafft adds : " Sed nullo artificio detegi potest terminus generalis." But this statement is very strange ; for if we put a for ^ , and ft for , it can be easily shewn that & the nth term of the series is a n + /3 W . The sum of n terms can then be readily obtained, and shewn to agree with the expression given above. The series occurs on page 7 of the Exercitationes Mathematicce, and there D. Bernoulli seems also to state that the general term cannot be expressed. He says : ...unicam seriem exempli loco afferam talem 1, 3, 4, 7, 11, 18, 29... in qua quilibet terminus duorum pneceileiitiuin est summa, et quuiu 444 MISCELLANEOUS INVESTIGATIONS BETWEEN 17G1 AND 1780. nunquam ad terminum generalem reduci posse demonstratum habeo ; Lsec series, quaruvis Geometris non coiisiderata, quod sciam, attentione tamen dignissima est ob multas, quibus gaudet, proprietates :... 695. Krafft's essay does not contribute in any way to the ad- vancement of the subject ; in fact the author by his ignorance of what had been effected by Maclaurin and by Simpson, shews that his knowledge was below the level it might have reached. The problem which we have noticed in Art. 690, may have been new at the time, but this is very uncertain. The address of the President, John Kies, to the author of the essay may be preserved as a specimen of the academical pleasantry of the last century : Gravissimum est argumentum, cujus elaborationem in Te snscepisti, miratus Tuum volatuni & Polo ad Aequatorem Te feliciter rediisse gratulor, calculorum quos evitare non licuit, neque multitude neque pondera Te a Tuo tramite potuerunt avertere, et uti in series eorum infinitas incidisti, ita inde reduci Tibi seriem prosperitatis et felicitatis iufinitam ex animo apprecor. Sequere porro vestigia Celeberrimi Parentis TOV /xa/<apiTov, Antecessoris niei in officio academico desideratissimi. Vale et me amare perge d. 8. Octobr. 1764. 696. We next notice a memoir entitled Pet. von Osterwald Bericht uber die wrgenommene Messung einer Grundlinie von Munchen bis Dachau.... This memoir is contained in the Abhandlungen der. . .Baierisclien Akademie...Vo\. II., 1764. It occupies pages 361... 386 of the volume. The base line to which this memoir relates may have been useful for the topography of Bavaria, but it has had I believe no influence on our subject. The base was measured twice; the first result was 43824 French feet; the second result was 10 feet 3 inches more. Five rods of fir-wood were employed, each 12 feet long. The temperature was higher on the average at the second measurement than at the first ; and to this circumstance Osterwald attributes the difference of the result. He gives an account of experiments to shew that heat contracted and cold expanded his MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 445 rods: but this seems very strange, and probably no one at the present day would accept such a conclusion. 697. In the Philosophical Transactions, Vol. LVL, which is for the year 1766, published in 1767, we have a memoir entitled Proposal of a Method for Measuring Degrees of Longitude upon Parallels of the ^Equator, by J. Michell, B.D., F.R.S. The memoir occupies pages 119... 125 of the volume: it was read May 8, 1766. Michell, in fact, proposed to measure an arc perpendicular to the meridian of a given place ; but he does not discuss the practi- cal difficulties which would occur in attempting to execute the design. The memoir indeed seems to belong to an earlier genera- tion, and to be quite out of date in 1766. 698. We have next to notice a memoir by Canterzanus, en- titled De Attractions Sphcerce. This memoir is contained in Vol. v., part 2, of the De Bononiensi...Academia Commentarii, published at Bologna in 1767; the memoir occupies pages 66... 70 of the volume. There is an account of the memoir on pages 175... 177 of the preliminary portion of Vol. V., part 1, of the same series ; this account is by Franciscus Maria Zanottus. Zanottus desired to give, as an appendix to a book on central forces, the theorem, that according to the ordinary law of attrac-. tion a sphere attracts an external particle, as if the mass of the sphere were collected at its centre. As in the book he had adopted short, simple, Cartesian explanations, Zanottus wished this theorem to be exhibited in like manner, or at least to be established by a strictly synthetical demonstration. It seems curious that at this time a book should have been written using Cartesian methods Cartesianos calcidos. Moreover, it is difficult to see why Zanottus could not be content with Newton's demonstration ; but to this he does not allude. Zanottus refers to a demonstration by Sigorgnius, which he condemns as unsatisfactory; and justly, assuming that he has reported it faithfully. 446 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. John Bernoulli, he says, had demonstrated the theorem, using infinitesimal differences, in the manner of Leibnitz. Gravesand also had given a demonstration, " brevem admodum, si tantum legas ; si comprehendere etiam veils, non admodum." Moreover, this demonstration was not really synthetical, but only an analytical one disguised. Finally Zanottus had recourse to Canterzanus, whom he de- scribes as "maximo ingenio juvenem ...... a quo nihil non speran- dum esse videbatur." Canterzanus accordingly satisfied him by a demonstration which occupies nearly five large pages. The de- monstration is sound, and not devoid of elegance. We will give a brief account of it, by the aid of algebraical symbols ; though this would probably have been very distasteful to Zanottus himself. Let A denote the centre of the sphere, any point outside the sphere ; we propose to find the attraction of the sphere at 0. Let OA = c ; let a denote the radius of the sphere. Let x lie in value between c a and c + a. Describe with as centre a spherical surface of radius x, and another of radius x + $x, where &x is infinitesimal. Between these two surfaces a portion of the given sphere is contained. The attraction of the portion at is along OA, and is ultimately equal to -2 x TTX Z sin 2 6S%, that is TT sin 2 where is the angle which a straight line drawn from to the boundary of the portion makes with OA : see Art. 8. This is, in fact, the essence of Canterzanus's method ; he ob- tains this result synthetically, and without difficulty. He then has to determine the whole attraction of the sphere ; this also he obtains synthetically, though with some little trouble. In modern notation we should say that the whole attraction is fc+a TT I sin 2 Odx ; and we should easily effect the integration by J ca observing that COS0 = The result will be MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 447 699. We now proceed to a memoir entitled Eustachii Zanotti. De angulo positionis et ejus usu in determinanda Telluris figura. This is published in the De Bononiensi Scientiarum...Commentarii. Vol. v. part 2, 1767. It occupies pages 256... 264 of the volume. Zanottus at Bologna observed the sun setting near a high tower of Modena. Hence he determined the angle of position of the tower by a way which he considered gave a very accurate result. Then if the difference of longitude between his observatory and this tower were known, he could calculate the difference of latitude on the assumption that the Earth is a sphere. By comparing this calculated difference with that assigned by direct observations, information would be obtained as to the Figure of the Earth. However, it is admitted, that practically the difficulty of fixing the difference of longitudes exactly renders the suggestion useless. Still Zanottus thinks that some advantages could be obtained by the use of the angle of position, when determined with the accu- racy which his method would ensure. 700. A memoir by J. A. Euler entitled, Versuch die Figur der Erden durch Beobachtungen des Monds zu bestimmen, is contained in the Abhandlungen der JBaierischen Akademie... Vol. V., Munich, 1768. The memoir occupies pages 199... 214 of the volume. The memoir consists of two parts. In the first part, assuming the form of the Earth, the influence exerted by this form on the meridian observations of the moon is investigated. In the second part it is proposed to determine the form of the Earth from such observations ; but the author himself admits that the process is not satisfactory. 701. A memoir by Lambert entitled Sur la Figure de I Oce'an is contained in the Berlin Memoires for 1767, published in 1769. The memoir occupies pages 20... 26 of the volume. The memoir is not mathematical, and does not belong to our subject, but rather to geology. 702. In Volume LVIII. of the Philosophical Transactions which is for 1768, published in 1769, we have an Extract of a Letter, dated Vienna April 4, 1767, from Father Joseph Liesganig, 448 MISCELLANEOUS INVESTIGATIONS BETWEEN 17C1 AND 1780. Jesuit, to Dr. Bevis, FM.S., containing a short Account of the Measurement of Three Degrees of Latitude under the Meridian of Vienna. This occupies pages 15 and 16 of the volume. It records the amplitudes and the lengths of the various parts into which Liesganig's entire arc was divided. See Art. 704. 703. In the same Volume of the Philosophical Transactions we have an account of the operations carried on by Charles Mason and .Jeremiah Dixon, for determining the length of a Degree of latitude in the provinces of Maryland and Pennsylvania, in North America. There is an introduction by Maskelyne, then the detail of the work by Mason and Dixon, and finally some remarks and a postscript by Maskelyne. The whole occupies pages 270... 328 of the volume. The peculiarity of these operations is that the whole length was actually measured with rods. I do not know what Maskelyne means by saying on page 325 that " an error of only 1" in the celestial measure would produce an error of no less than 67 feet in the length of the degree," Instead of 67 feet, it seems to me that we should read 100 feet, for the length of the degree is found to be more than 360000 feet : see his page 324. Maskelyne himself had supposed that from the nature of the country no deflections of the plumb-line were to be feared ; then he says on his page 328 : But the Honourable Mr. Henry Cavendish lias since con- sidered this matter more minutely, and having mathematically inves- tigated several rules for finding the attraction of the inequalities of the Earth, has, upon probable suppositions of the distance and height of the Allegany mountains from the degree measured, and the depth and declivity of the Atlantic ocean, computed what alteration might be produced in the length of the degree, from the attraction of the said hills, and the defect of attraction of the Atlantic; and finds the de- gree may have been diminished by 60 or 100 toises from these causes. He has also found, by similar calculations, that the degrees measured in Italy, and at the Cape of Good Hope, may be very sensibly affected by the attraction of hills, and defect of the attraction of the Mediterranean Sea and Indian Ocean. MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 449 Frisi, in his Cosmograpkia, Vol. II., page 91, seems to think that this diminution of 60 or 100 toises in the American arc can hardly be accepted. It is indeed difficult to see how the Atlantic ocean can have produced any appreciable effect. Bailly refers to the passage without any expression of dissent: see his Histoire de I Astronomic Moderne, Vol. in., page 41. On pages 329... 335 of the same volume of the Philosophical Transactions, some pendulum observations are recorded, which were made by Mason and Dixon at the northern end of their arc, that is in latitude 39 56' 19" North. 704. An account of the measurements of arcs of the meridian in Austria and Hungary by Liesganig, was published at Vienna in 1770, under the title of Dimensio Graduum Meridiani Viennensis et Hungarici. The volume is in quarto; it contains a Dedication and an Introduction, the text on 262 pages, and a leaf of Errata; there are ten plates. The volume contains no theoretical investigations, so that it does not full within our range. Practically the results of the operations do not seem to be esteemed of any value: see De Zach's Corres- pondance Astronomique, Vol. VII. ; and the article Figure of the Earth in the Encyclopaedia Metropolitana, page 170. 705. In the Philosophical Transactions^ 'ol. LXL, for 1771, pub- lished in 1772, we have a memoir entitled An attempt to explain some of the principal Phenomena of Electricity, by Means of an elastic Fluid: By the Honourable Henry Cavendish, F.R.S. The me- moir occupies pages 584... 677 of the volume. This memoir would require careful attention in a History of Electricity; but a very brief notice will suffice for our purpose, as it contributes nothing that is really new to the theory of attraction. We have, on pages 586 and 587, the attraction of a cone on a particle at the vertex, assuming the law to be that of the inverse wth power of the distance. We have, on page 592, the enunciation of the proposition that, on the ordinary law of attraction or repulsion, a spherical shell does not exert any action on an internal particle: 'for the demon- stration we are referred to Newton's Pi incipia, Lib. i., prop. 70. T. M. A. 29 450 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. Cavendish adds : It follows also from his demonstration, that if the repulsion is in- versely, as some higher power of the distance than the square, the particle P will be impelled towards the center; and if the repulsion is inversely as some lower power than the square, it will be impelled from the center. Hence, if the law of attraction or of repulsion is given to be that of some single power of the distance it follows that a particle will not be at rest when placed inside a spherical shell, unless the law be that of the inverse square. But this does not apply if the law is merely assumed to be expressed by some function of the distance, as for example by hr m + kr n , where r denotes the distance, and the other letters denote constants. The general proposition was given by Laplace in the Mecanique Celeste, Livre II. 12 : it has since passed into the elementary treatises on Attraction. On Cavendish's page 616, we have an investigation of the attraction of a circular lamina on an external particle symmetri- cally situated; the expression obtained is made to yield various easy inferences in the subsequent pages. 706. In the Paris Memoir es for 1772, Second* partie, published in 1776, there is a memoir by La Condamine, entitled Bemarques sur la Toise-tialon du Chdtelet, et sur les diver ses Toises employees aux mesures des Degres terrestres et a celle du Pendule a secondes. The memoir occupies pages 482... 501 of the volume; see also pages 8... 13 of the historical portion of the volume. The memoir was read on the 29th of July, 1758 ; but was forgotten by its author, and found two years after his death. The object of the memoir is to recommend the toise of the Equator as the standard toise ; this toise is that which was used in measuring the arc of the meridian in Peru, and which is elsewhere called the toise of Peru: see Arts. 186 and 551. A certain iron toise existed which was theoretically the stand- ard toise; this was the Toise-^talon du Chdtelet, so called from the place where it was kept. This standard appears to have been fixed in its place in a wall in 1668; and Picard adjusted by it the toise which he used for measuring his arc between Paris and Amiens. At the date of the composition of La Condamine's MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 451 memoir, the standard was damaged and no longer trustworthy; and Picard's toise had not been preserved. La Condamine gives information respecting the toise of the Equator; and he compares with it various other toises, beginning with that used to measure the arc in Lapland, called the toise of the North. With respect to the toise of the Equator and the toise of the North La Condamine says on his page 492: II est vrai que depuis que les deux toises sont revenues en France, on a cru trouver entr'elles, par une nouvelle comparaison, une legere difference, qu'on a juge"e d'un vingtieme ou d'un trentieme de ligne (dont la toise du Nord est plus courte) en attendant une determination plus precise. Voyez le rapport des quatre Commissaires, insere dans les Memoires de F Academic de Pannee 1754, p. 178; et le Journal des operations de M. le Moiinier, imprime au Louvre en 1757, page 8, ligne 11. We have alluded to this in Art. 551. La Condamine thinks that the toise of the Equator and the toise of the North were originally of the same length, and that the slight difference between them arose from the shipwreck in the Gulf of Bothnia when the expedition returned from Lapland. At a later period, when these two toises were again compared, they were found to be equal: see the Base du Systeme M&rique, Vol. in., page 413. La Condamine considers in the next place the toise which was used in the geodetical operations in France in 1739 and 1740, and in 1756; this toise is called the toise of the Observatory or of the Degrees of France. La Condamine arrives at the conclusion that this toise was practically equivalent in length to the toise of the Equator. La Condamine also thinks that the toise used by La Caille for measuring an arc at the Cape of Good Hope, agrees with the toise of the Equator. La Condamine refers also to a toise which had been used by De Mairan in some pendulum experiments: La Condamine con- siders that this toise is about a tenth of a line shorter than the toise of the Equator. But I have found a statement by D'Alembert 292 452 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. which is not quite consistent with this; it is in the article Figure de la Terre in the original Encyclopedic, page 754: or la toise de M. de Mairaii est aussi la meme qui a servi a la mesure des degres sous 1'equateur et sous le cercle polaire, et la meme qu'on a employee pour v6rifier en 1740 la base de M, Picard. I presume that the authority of La Condamine is superior to that of D'Alembert. La Condamine briefly examines the various methods of pre- serving an exact record of the standard of length: he recommends that the standard should be hollowed out in a table of porphyry or of granite. In a note at the end of the Memoir, we are told that the pro- position to take the toise of the Equator as the standard toise was not adopted in 1758, owing to the opposition of De Mairan; but in 1766 the royal authority was exerted in favour of it. The memoir is of great importance with respect to standards of length ; it contains references on the subject to memoirs in the preceding volumes of the Paris Academy, and to other works. An interesting paragraph respecting a suitable universal standard of length occurs on page 500 ; it begins thus : M. Mouton, Chanoine de Lyon, est le premier, que je sache, qui proposa cette mesure tiree du pendule; ce fut en 1670. Observations diam. Sol, Lun. Lyon, publiees en 1670. 707. A memoir by Lagrange entitled Sur r attraction des sphe'roides elliptiques is contained in the Nouveaux Mtmoires de VAcadtfmie... Berlin for 1773, published in 1775. The memoir occupies pages 121... 148 of the volume. 708. Lagrange refers to the investigations given by Maclaurin in his Prize Essay on the Tides. Lagrange says on his page 121 : ... il faut avouer que cette partie de 1'Ouvrage de M. Maclaurin est un chef-d'oeuvre de Geometric, qu'on peut comparer a toi& ce qu'Ar- chimede nous a laisse de plus beau, et de plus ingenieux. Lagrange proposes to demonstrate by analytical methods the results which Maclaurin demonstrated by geometry. MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 453 709. Lagrange speaks thus of Simpson on his page 122 : On trouve a la v6rite dans les Ouvrages de M. Thomas Simpson tine solution purement analytique du probleme de M. Maclaurin, dans laquelle on ne suppose point que le spheroi'de elliptique soit a tres peu pres spherique; mais d'un autre cote cette solution a le defaut de proceder par le moyen des series, ce qui la rend non seulement longue et compliquee, rnais encore peu directe et peu rigoureuse. I suppose that the defect which Lagrange has'in view when he describes Simpson's solution as deficient in rigour is the fact that the series are not always convergent : see Art. 282. 710. The general formula for the attraction of a body on a particle in terms of rectangular coordinates are first investigated, and it is remarked that the great difficulty is to effect the integra- tions which are indicated in the formulae. This leads to the sub- ject of the transformation of the variables in a triple integral. Lagr-ange gives the method which has since remained in nearly all our elementary books ; although obscure and unsatisfactory. I have supplied an account of this method, and indicated its defects, in my Integral Calculus, where I have explained and adopted another method. The transformation of multiple integrals is an important branch of analysis ; we may consider that Lagrange was the author of it, and that the subject of attraction suggested the consideration of it to him. See Lacroix Traiti du Calcul Differentiel et du Calcul Integral, Vol. n., page 208. As an example of this transformation, Lagrange gives the for- mula, now very familiar, by which we pass from rectangular to polar coordinates in the expression for an element of volume, namely dxdydz = r 2 sin 6d0d<j>dr. 711. Lagrange works out the case of the attraction of an oblatum on an internal particle ; the process is essentially the same as would be found in any modern treatise on attractions : see, for example, titatics, Chapter xili. 454 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 712. Lagrange says on his page 139 : M. Maclaurin dans son Traite du flux et du reflux de la mer s'est content^ de chercher 1'attraction d'un spheroide elliptique sur un point quelconque de ce spheroide; et les resultats de sa belle rnetliode synthetique s'accordent parfaitement avec ceux quo nous venons de trouver par ]' Analyse. M. d'Alembert vient d'etendre la solution de M. Maclaurin a des spheroi'des ou toutes les coupes seroient elliptiques, en faisant remarquer que les propositions qui servent de base a cette solution sont egalement vraies a 1'egard de tous les spheroi'des elliptiques, soit de revolution ou non; c'est ce que nous avons trouve directement par notre Analyse... The researches of D'Alembert which are here noticed are, I presume, those in the sixth volume of the Opuscules Mathema- tigues. In the last words of the preceding extract, Lagrange alludes to the demonstration of various properties in the attraction of an ellipsoid, not necessarily of revolution, on an internal particle. Lagrange shews: that as long as we keep on the same radius drawn from the centre of the ellipsoid the attraction varies as the distance from the centre ; that the attraction of a shell bounded by similar, similarly situated, and concentric ellipsoidal surfaces on a particle within the shell is zero ; and that the attraction resolved parallel to an axis varies as the perpendicular distance from the plane which contains the other axes. For the case of an ellipsoid of revolution these results had been long known ; the first and the second were given by Newton, and the third by Maclaurin. It had more recently been shewn that they were also true for ellip- soids not of revolution : see Arts. 615 and 662. Lagrange now establishes these propositions by analysis. 713. With respect to the absolute value of the attraction of an ellipsoid on an internal particle, Lagrange says on his page 139: ...a Tegard de la valeur absolue de 1'attraction des sphe*roi'des qui ne sont pas de revolution, M. d'Alembert a essayo de la determiner par differens moyens tres ingenieux, mais dont aucun ne lui a pleinement reussi ;... MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 455 714. Lagrange also investigates the attraction of an oblatum on an external particle which is situated on the prolongation of the axis of revolution. He says at the end of the investigation on his page 144: Ce Probleme a aussi ete resolu synthetiquernent par M. Maclaurin dans son Traite des fluxions, et nos solutions s' accordant dans les re- sultats. In the course of the investigation Lagrange allows himself to fall under the suspicion of contradicting the first principles of the Integral Calculus : see his pages 142 and 143. He says in fact that iPdp vanishes when taken between the limits a and a, where P is a function of p which is always positive. The result at which he arrives is correct, but his method is unsatisfactory. Instead of integrating with respect to p between the limits a and a, and then with respect to q between the limits and TT, he ought to integrate with respect to p between the limits and a, and with respect to q between the limits and 2?r. His polar expression for the element of volume really assumes that sin^? is always positive. 715. Lagrange alludes to the case of the attraction of an ellipsoid, not of revolution, on an external particle which is situated on the prolongation of an axis. He says on his page 145 : mais Tiiitegration de la differentielle dont il s'agit e"tant tres difficile, si meme elle n'est pas impossible, nous ne nous y arreterous pas; outre que cette matiere n'est pas proprement de 1'objet auquel ce Memoire e*toit destine, elle a d'ailleurs e"te dej savamment diseutee dans le sixieme Volume des Opuscules deM. d'Alembert, auquelil nous suffira par consequent de renvoyer. Lagrange says {hat we should find still greater difficulties in attempting to investigate the attraction of an ellipsoid on any external point. He shews, what the expressions which have to be integrated become when the axes of coordinates are shifted so that one of them is made to pass through the external point. 456 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 716. The memoir does not proceed so far in the subject as Maclaurin's Treatise of Fluxions did ; for the theorem which we have reproduced in Arts. 257 and 258 is not demonstrated by Lagrange. But as we shall see Lagrange added the demonstra- tion in the Berlin M&noires for 1775. 717. An account of a measurement of an arc of the meridian in Lombardy by Beccaria was published at Turin in 1744 ; I have not seen the volume. The result obtained by this operation appears however never to have been received with confidence. See the memoir by De Zach in the Turin M6moires for 1811 and 1812; and De Zach's Correspondence Astronomique, Vol. vn., page 502 ; and also the article Figure of the Earth in the Ency- clopedia Metropolitana, pages 170, 208, and 210. 718. A work was published in Florence in 1777, entitled Lettere di un Italiano ad un Parigino intorno alle riflessioni del Sig. Cassini de Thury sul grado Torinese. The work consists of 67 octavo pages : it is anonymous. Cassini de Thury seems to have made some remarks on Beccaria's book in the Mercure de France for 1776 ; and the present work is a reply. The main purport of the reply is to shew that Beccaria's result was what might have been expected if due allowance were made for the attraction exerted by the Alps. There is however no theory nor calculation in the book, but only general considerations. I have not seen the remarks which Cassini de Thury made ; but judging from the reply, they contained some inaccuracies or misprints. 719. We have referred in Art. 643 to the extracts from two letters which D'Alembert addressed to Lagrange ; the letters gave rise to some investigations by Lagrange, which we shall now notice. 720. In the Nouveaux Mtmoires de rAcade'mie...~BeT\m for 1775, published in 1777 we have on pages 273... 279 an Addition au Mdmoire sur ^attraction des sphe'ro'ides elliptiques imprime' dans le Volume pour lAnnfe 1773, par M. de La Grange. MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 457 This addition was read on the 9th of November 1775 ; it com- mences thus : Les reraarques contenues dans la lettre de M. d'Alembert dont j'ai eu 1'homieur de faire part a 1' Academic il y a huit jours, m'out donne occasion de chercher si le Theoreme de M. Maclaurin concernant 1'attraction d'un ellipsoide sur un point quelconque place dans le prolongement de 1'un de ses trois axes ne pourroit pas se deduire des formules que j'ai donnees dans ce Memoire; et jecrois que les Analystes verront avec plaisir avec combien de facilite on pent parvenir par ces formules a la demonstration du Theoreme dont il s'agit. 721. Lagrange starts with a formula which he had obtained in the memoir of 1773 for the attraction of an ellipsoid on a point on the prolongation of an axis. Let c be the polar semiaxis, a the equatorial semiaxis, of an oblatum; let the density be unity. Then for the attraction on a particle on the prolongation of the polar axis at the distance r from the centre we have by Art. 261, the expression __ V( 2 -c 2 ) C 2 Put m for -3 ; then the expression becomes Now let us suppose that instead of an oblatum we have an ellipsoid ; let a and b be the semiaxes of the section which is at right angles to the distance r\ and let c as before be the semiaxis in the direction of r. The attraction will be equal to the integral between and 2?r of tVm V((l-m)cl M l] M J Zir 1 where m now denotes This is obvious ; for a wedge of this ellipsoid made by two planes inclined respectively at angles and 6 + dd to the plane 458 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. of a and c may be considered as equivalent to the wedge of an oblatum which has for semiaxes c and a l9 where flj'cos'0 a* sin g fl ~~~~ ~~ so that a 2 = -c-r a so Thus from the known value of the attraction of an oblatum we have deduced the expression which must be integrated in order to determine the attraction of an ellipsoid in the case under consider- ation. The method we have used is not formally identical with Lagrange's, but it is coincident in principle. Lagrange then by a suitable transformation of the expression just obtained, succeeds in demonstrating Maclaurin's theorem. 722. Strictly speaking, Lagrange's demonstration applies only to the case in which the attracted particle is on the prolongation of the least axis of the ellipsoid. But.it will be found on exami- nation, that the method may be applied with obvious modifications to the cases of the other axes. Lagrange finishes thus: C'est le theoreme que M. Maclaurin a 6nonce sans demonstration dans 1'Art. 653 de son Traite des fluxions; et que nous nous etions propose de deduire de nos formules. As we have already stated in Art. 260, Lagrange underrates what Maclaurin really effected. 723. A work was published in 1775, by Cassini de Thury, entitled Relation dun Voyage en AUemagne,...Suivie de la Descrip- tion des Conquetes de Louis XV, depuis 1745 jusqu? en 1748. This work is in quarto, containing xxviii + 194 pages: it may have a brief notice, though very slightly connected with our subject. Cassini de Thury travelled to Vienna, and made numerous obser- vations of angles in the course of his journey. He calculated a large number of distances, by means of triangles; and the data and the results are recorded. There are numerous maps on which MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 459 the triangles are drawn. On pages 1...6, we have an account of the measurement of a base near Munich. On page xx. there is an allusion to an Observatory which Cassini had formerly constructed on the top of a tree 100 feet high : it seems to be the same as we noticed in Art. 226. Cassini de Thury was present with the French army during part of the war in Flanders in 1745 and 1746. He records a set of triangles, and gives a map, extending over the range of the French conquests. He says on page 124: telle est 1'iclee que Ton doit se former de 1'etendue des con- quetes du feu Roi, que j'ai tache de representer dans une Carte generale r qui est le seul monument qui nous en reste, si Ton compte pour rien une longue paix qui en a ete la suite, et que le plus aime des Hois preferoit a la victoire. Cassini de Thury refers on page iii. to a work published in 1765; this I have not seen, but conjecture to be that of which the title is given on page 483 of La Lande's Bibliographie Astrono- mique with the date 1763. At the end of the volume we have, on five pages, Extraits des Registres de VAcademie. . . : these furnish an account of the book, by Laplace, signed by Le Monnier and himself. The work cannot be considered of any scientific importance: see the Paris Mdmoires for 1775, pages 41... 44 of the historical por- tion ; and De Zach's Monatliche Correspondents, Vol. vn., page 397. 724. In the Philosophical Transactions, Vol. LXV., for 1775, published in 1775, there are two memoirs by Maskelyne. The first memoir is entitled A Proposal for measuring the Attraction of some Hill in this Kingdom by Astronomical Observa- tions. This occupies pages 49 5... 499 of the volume : it was read in 1772. The second memoir is entitled An Account of Observations made on the Mountain Schehallien for finding its Attraction. This occupies pages 500... 542 of the. volume: it was read July 6, 1775. 460 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. Mr Charles Mason examined various hills in England and Scotland, and selected Schehallien, in Perthshire, as suitable for the proposed operations. Maskelyne's second memoir details the astronomical and geodetical proceedings, and records the observa- tions of the stars. From a preliminary determination founded on observations of 10 stars it appeared that a deviation of 11"' 6 was produced in the plumb-line by the sum of the attractions on the North and South sides of the mountain. 725. The following remarks occur in the first memoir, on page 496 : Sir ISAAC NEWTON gives us the first hint of such an attempt, in his popular Treatise of the System of the World, where he remarks, "That a mountain of ail hemispherical figure, three miles high and six broad, will not, by its attraction, draw the plumb-line two minutes out of the perpendicular." It will appear, by a very easy calculation, that such a mountain would attract the plumb-line V 18" from the perpendicular. The work to which Maskelyne here alludes is entitled A Treatise of the system of the World. By Sir Isaac Newton. London, 1728. This purports to be a translation of the popular exposition drawn urj by Newton himself, to which he refers at the beginning of the third Book of the Principia. The date 1728 is after the death of Newton. The passage which Maskelyne quotes is from page 41. On the same page is a statement equivalent to that which we have noticed in Art. 125, so that Maupertuis must bave taken it from this book. Since that Article was printed, I have obtained a copy of the second edition of Maupertuis's Figure des Astres ; the statement is omitted in this edition. The passage stands thus : ...For the attractions of homogeneous spheres near their surfaces, are as their diameters. Whence a sphere of one foot in diameter, and of a like nature to the Earth, would attract a small body plac'd near its surface, with a force about 20000000 times less, than the Earth would do if placed near its surface. But so small a force could produce no sensible effect. If two such spheres were distant by j of an inch, they would not even in spaces void of resistance, come together by the MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 461 force of their mutual attraction in less than a months time. And less spheres will come together at a rate yet slower, viz. in the proportion of their diameters. Since the statement is ascribed to Newton it may be proper to give an investigation, which need not appear necessary when only the authority of Maupertuis was involved. Let a denote the radius of each sphere, R that of the earth, 2a -+ c the original distance of the centres of the spheres. When the centres of the spheres are at the distance r the acceleration which tends to bring them nearer is 2 -= g . I - j . Thus during the motion this acceleration lies between -~ { ) , and R \2a + c/ -jj- ( 5~ ) If the acceleration were constant and equal to f, the time of motion would be */ . Hence the real time lies between ag In the example it is said that the sphere is one foot in diameter, but this must be a mistake for one foot in radius. Thus a = 1, c = ^ , g = 32, and R = 20000000 ; therefore the time in / 1 \ / ^ \ 4 / ^ \4 seconds is between ( 2 + TT] 1000 [^, J and 2000 ( - ) , that is less than 250 seconds. This differs so widely from what we find in the foregoing passage, that the words must I conclude have some meaning different from that which they appear to^suggest. The preceding elementary considerations are sufficient for our purpose ; but there is no difficulty in supplying an exact investi- gation. Let x denote the distance of the centre of one sphere from a fixed origin at the instant denoted by t, and x the distance of the centre of the other sphere ; suppose x greater than x, and 462 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. put r for x x. Let m denote the mass of each sphere, M the mass of the earth, and T the whole time of motion. Then d*x _ m d*x _ m di?~(x-x)*' ~df = ~(aT-x} z ' hence by subtraction m therefore [ ~ ] = 4m ( T ) , \dtj \r bj where b denotes the initial value of r, that is 2a -f c. Therefore T^ \" ,^-*L ^mJ 2a ^(b r) A , m _m M IP _a tf-M'tf'^-ti 9 ' Assume r = b sin 2 ^ ; thus where sm*/3 = -j-. This is exact, and the value may be easily calculated nu- merically. Suppose c small compared with b ; then we may approximate thus: Put 7 for - /3, therefore sin 2/3 sin 2y = 2y nearly. Jl Also cos 7 = A/~A~ == ( ~jT~)i so ^ na ^ a PP rox i ma tely Hence MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 463 726. Maskelyne obtained some results by calculation, which are thus stated : By calculation... it should follow, that the sum of the contrary attractions of Whernside...on the plumb-line placed half-way up the hill, would not be less than 30", and might amount to 46"... By a calculation..., the sum of the contrary attractions of the plumb- line, placed alternately on the North-side of Helwellin, and the South- side of Skidda, amounts to about 20"... And although the density of the earth near the surface should be five times less than the mean density, as there is some reason to suspect, and the attractions, as here stated, should consequently be diminished in the proportion of five to one, still the sum of the contraiy attractions of Whernside would be 6" or 9", and the sum of the contrary attractions of Helwellin and Skidda would be 4" . . . 727. On the whole 43 stars were observed, and 337 observa- tions taken. Maskelyne proposed at his leisure to compute the result from all the observations : see his page 530. It does not appear tbat this was done by him ; but it was by De Zach ; see L? attraction des Montagues, pages 686... 692. The result obtained by De Zach agrees very closely with Maskelyne's own. Some important calculations were founded on Maskelyne's result by Hutton in 1778, and by Playfair in 1811 ; these we shall consider hereafter : see Art. 730. 728. We next notice a work entitled Essai sur les Phenom&nes relatifs aux disparitions ptfriodiques de VAnneau de Saturne. Par M. Dionis da Sejour. Paris, 1776. This is an octavo, and contains xxxii + 444 pages, besides title-pages, and plate. "A notice of the work is given in the historical portion of the Paris Memoir es for 1775, pages 5 3... 5 5. The work relates to the appearances presented by Saturn's ring, and barely touches on the theory with which we are concerned. On pages 402... 406 we have the equation which Maupertuis investigated for the form of the ring ; for the demonstration we are referred to Maupertuis : see Art. 119. 464 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. On pages 407... 411 we have a formula for the attraction of a circular lamina on a constituent particle : the formula however would not be of any use, because the expression to be integrated becomes infinite within the range of integration. The author believes that the parts of the ring of Saturn must be animated by a centrifugal force in order to balance the effect of the attraction of Saturn : see his pages iv and 401. This supposition has been confirmed since by the researches of Laplace, and the observations of Herschel. 729. A work was published by John Whitehurst entitled An Inquiry into the original state and formation of the Earth... The first edition appeared in 1778, and the second in 1786 ; both are in quarto. In Button's Philosophical and Mathematical Dictionary under the head Whitehurst, it is stated that a third edition appeared in 1792. The work is geological, and not mathematical, and so does not fall within our range. I extract one sentence which reproduces an undemonstrated assertion noticed in Art. 130 ; it occurs on page 6 of the first edition: "...and therefore when the component parts of fluid bodies are thus assembled together, they must necessarily assume spherical forms...." 730. In the Philosophical Transactions, Vol. LXVIIL, for 1778, part 2, published in 1779, there is a memoir by Hutton, entitled An account of the Calculations made from the Survey and Measures taken at Schehallien, in order to ascertain the mean Density of the Earth. The memoir occupies pages 689... 788 of the volume : it was read on May 21, 1778. The attraction of the mountain had to be calculated for each of the two stations at which Maskelyne made his astronomical observations. The form of the mountain was ascertained by a very minute survey ; then it was supposed to be decomposed into slender vertical prisms, the attraction of every one of which could be calculated. There were 960 such prisms for each of the two stations. MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 465 The numerical labour was of course very great ; but the memoir adds nothing to the theory of Attraction. We may notice the method adopted for facilitating the calculation, derived, as Hutton says on his page 750, " partly from some hints of the Honourable Henry Cavendish, F.R.S. and partly from some of my own, which had been communicated to the Astronomer Royal in the years 1774 and 1775..." Compare the note on page 237 of the article, Figure of the Earth in the Encyclopaedia Metropolitans. Take the horizontal plane through one of the stations for the plane of polar coordinates ; let the station be the origin, and let the initial line be in the direction of the meridian. Let there be a vertical prism standing on a base which has r and 6 for the polar coordinates of a corner ; and let r Ar A# denote the area of the base. Suppose z the height of the prism ; then, taking the density as unity, the horizontal attraction of the prism is very approximately // 2 , ' g\ and the resolved part of this in z ) the direction of the meridian is -r-*, -K cos 6. This may be put in the form -? r (sin (6 + A0) - sin 6} Ar. Then to facilitate the calculation the values of A# were so taken as to make sin (6 + A0) sin Q retain a constant value. Thus, to obtain the attraction of the prisms forming part of the same ring, the values f ~77~2 - 2\ must be summed, and the result obtained must be multiplied by {sin (6 + A0) - sin 6} Ar. Button's conclusion is that the mean density of the Earth is o, K of that of the mountain. He conjectures that the mean density o of the mountain may be times that of water ; so that the mean 9 density of the Earth is about = times that of water. See Art. 17. Button's memoir is reproduced in his Tracts on Mathematical and Philosophical Subjects, Vol. II. 1812, with the addition of a few remarks towards the end, on the character of the mountain, which T. M. A. 30 466 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. were derived partly from Mr Duncan Macara, and partly from Professor Playfair. 731. In the Philosophical Transactions for 1811, part 2, pub- lished in 1811, there is a memoir by Playfair entitled Account of a Lithological Survey of Schehallien, made in order to determine the specific Gravity of the Rocks which compose that Mountain. The memoir occupies pages 347... 377 of the volume: it was read June 27, 1811. It will be convenient to notice this memoir here. Hutton, as we have seen, calculated the attraction of Schehallien, and thence deduced the mean density of the Earth, on the suppo- sition that the mountain was homogeneous ; and he assumed 2'5 for the density, that of water being unity. Playfair investigated the composition of the mountain, and modified the calculations by allowing for the actual density of the parts. Playfair found that the upper part of the mountain was composed of quartz of the mean specific gravity 2*6398 ; and that the lower part was com- posed of mica and hornblend slate of the mean specific gravity 2-83255, and limestone of the mean specific gravitj 7 - 276607. On the whole he considered that the matter composing the mountain could be divided into two classes of rocks ; namely, quartz of the mean specific gravity 2'639876, and micaceous rock, including cal- careous, of the mean specific gravity 2 '81 039. The line separating the two classes of rocks could be accurately traced on the face of the mountain. As to the arrangements in the interior of the mountain, Playfair considered that only two suppositions could be made with any degree of probability ; these amount to assuming that the two classes of rocks are separated by a vertical boundary, or by a nearly horizontal boundary. Playfair calculates the mean density of the Earth on both suppositions ; on the former he obtains 4'55886, and on the latter 4'866997. The memoir adds nothing to the theory of Attraction. Playfair availed himself of the practical method for facilitating the compu- tation which is given in Hutton's memoir. Playfair says, on his page 364 : I have also used a theorem in these computations, which gives an accurate value of the attraction of a half cylinder of any altitude a, MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 4G7 and any radius r, on a point in the centre of its base, and in the direc- tion of a line bisecting the base. Let A be equal to that attraction ; then The investigation of this formula may be usefully supplied. Take the origin at the attracted point ; let the axis of z coin- cide with the axis of the cylinder, and the axis of x with the direction in which the attraction is estimated. Then the resolved attraction is xdxdydz /7f JJJ We integrate first with respect to z, between the limits and a ; thus we obtain axdxdy The limits for y are V(^ ^ 2 ) and *J(v* x 2 ) ; and the limits for x are and r. Assume x 8 cos 9 and y = s sin ^ thus the integral trans- forms into and the value is 2 log There is something wrong about the plates which ought to accompany the memoir. Playfair refers to a plan of the mountain which shews the boundary between the two classes of rocks, and also to a diagram ; see his page 363 : but neither of these is given. Also it appears from his page 365, that the plates which are given ought to have been coloured. 732. It will be convenient to notice here a subsequent paper connected with the Schehallien experiment In the Philosophical 302 458 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. Transactions for 1821, part 2, published in 1821, there is a memoir by Hutton, entitled On the mean density of the Earth. It occupies pages 276... 292 of the volume : it was read April 5, 1821. Hutton adverts to the Schehallien operations, and to Playfair's investigation x>f the density of the mountain. Hutton thinks that the mean density of the Earth is very nearly five times that of water, but not greater. He prefers the Schehallien determination to that which Cavendish had obtained by experiment. At the age of 84, he had undertaken to recompute the experiments of Cavendish, and had discovered some important errors in the origi- nal computation. He suggests that observations might be made near one of the Egyptian pyramids, of the nature of those made at Schehallien. 733. We will collect here the titles of some investigations which deserve to be studied by those who are interested in the important question of the mean density of the Earth ; the greater part of these investigations fall without the period over which the present history ranges. We suppose the density of water to be unity. A famous experiment was made by Cavendish, from which he deduced that the mean density of the Earth was about 5'48. The details are given in the Philosophical Transactions for 1798 ; we shall recur to the memoir hereafter. A memoir by Hutton in the Philosophical Transactions for 1821 we noticed in Art. 732. According to Button's calculation the result of Cavendish's own experiments is 5*31. A paper by Carlini in the Milan Ephemeris for 1824 gives an account of a series of pendulum experiments made at the height of a thousand toises. The result obtained for the mean density is 4*39 ; but a serious error is introduced by using a wrong formula to express a certain attraction ; the error was pointed out by Schmidt and by Giulio. Moreover there are other considera- tions which shew that the process does not seem to deserve much confidence. See the Report on Astronomy by the Astronomer Royal in the Reports of the British Association, Vol. i. page 169; and a note by Sabine in the translation of Humboldt's Cosmos, Vol. I. 1849, page xlvii. MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 469 The subject is discussed in Schmidt's Lehrbuch der mathema- tischen und physischen Geographic, 1830; see Vol. II. pages 469... 487. Schmidt corrects Carlini's result 4*39 to 4'837. Schmidt ob- tains 5 '5 2 by calculation from Cavendish's own experiments. In 1838, a work was published at Freiberg, entitled Versuche ilber die mittlere Dichtigkeit der Erde...von F. Reich. In his introduction he refers to an article by Muncke in the new edition of Gehler's Physikalisches Worterbuch Vol. ill. page 940..., as giving a valuable comparison of the results hitherto obtained. Reich himself repeated Cavendish's experiment and "obtained nearly 5*44 for the mean density. A memoir by Menabrea is published in the Turin Memorie, Vol. II. 1840, entitled Calcul de la densite de la Terre. This con- tains an investigation of the theory connected with Cavendish's experiment. A memoir by Giulio is published in the Turin Memorie, Vol. II. 1840, entitled Sur la determination de la densite" moyenne de la terre, deduite de I 'observation du pendule faite d Ihospice du Mont Cenis par M. Carlini en Sept. 1821. Carlini's result 4'39 is here corrected to 4'95. Cavendish's experiment was repeated by Baily, who made far more trials, and with greater precautions, than his predeces- sors. The details form Vol. XIV. of the Memoirs of the Royal Astronomical Society, 1843; see also the references connected with this volume in the Royal Society's Catalogue of Scientific Papers under the head Baily, No. 45. The result obtained for the mean density was about 5*67; but the results of individual experiments were found to vary considerably. In the Philosophical Transactions for 1847 there is a memoir by Hearn entitled, On the cause of the discrepancies observed by Mr. Baily with the Cavendish apparatus for determining the mean density of the Earth. The discrepancies are attributed to the influence of mag- netism. 470 MISCELLANEOUS INVESTIGATIONS BETWEEN 17G1 AND 1780. After the publication of Baily's result Reich again repeated the experiment : see the Leipzig Abhandlungen, Vol. i. 1852, and the Royal Society's Catalogue of Scientific Papers under the head Reich, No. 17. 'Ihe result gives about 5'58 for the mean density. Reich refers to Hearn's memoir, but does not agree with it. In the Account of the... Principal Trianyidations...l858, which forms part of the Ordnance Survey of Great -Britain... the subject is discussed in Section x. Some account is given of preceding researches, together with the details of a new operation, like that at Schehallien, on the hill called Arthurs Seat at Edinburgh. The new operation gives the mean density 5*316. Pendulum experiments were made in 1854, by the Astronomer Royal in Harton Colliery for ascertaining the mean density of the Earth : see the Philosophical Transactions for 1856, and the Royal Society's Catalogue of Scientific Papers under the head Airy, Nos. 100, 101, and 110. The result is the value 6'566. The Astronomer Royal observes with respect to this result on page 342 of the Philosophical Transactions for 1856: The value thus obtained is much larger than that obtained from the Schehallien experiment, and considerably larger than the mean found by Baily from the torsion-rod experiments. It is extremely difficult to assign with precision the causes or the measures of the error of any of these determinations; and I shall content myself with expressing my opinion, that the value now presented is entitled to compete with the others, on, at least, equal terms. Haughton, in the Philosophical Magazine for July, 1856, by a special method deduces the result 5'48 from the Harton Colliery experiments. A memoir was published in Gottingen, in 1869, entitled Ueber die Bestimmung der mittleren Dichtigkeit der Erde von Anton Schell This is in quarto, containing 39 pages, with three plates. It is a useful account of various researches on the subject. MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. 471 734. We notice next a memoir entitled De Figura Terrce Commentatio. Autore J. A.J. Cousin, Parisino. This is contained in the Acta Academics Electoralis Moguntince...l777. Erfurti. 1778. It occupies pages 209... 216 of the volume. The memoir consists chiefly of relations between certain lines in any curve, expressed in the language of the Differential Calcu- lus. But no diagram is supplied, at least in the only copy which I have seen, and thus part of the memoir is unintelligible. How- ever it may be safely pronounced to be of no importance. 735. A memoir by Euler, entitled Theoria Parallaxeos ad Figuram Terrce sphaeroidicam accommodata, is contained in the Acta Academice...Petropolitance for 1779, pars prior, published in 1782. The memoir occupies pages 241... 278 of the volume. This memoir adds nothing to the theory of the Figure of the Earth. Euler assumes that the Earth is an oblatum, and investigates the consequent expressions for the moon's parallax. He gives tables and numerical examples, which are calculated on the supposition that the ellipticity is -==: . 736. In the Philosophical Transactions for 1780, published in 1780, there is a memoir by Hutton, entitled Calculations to deter- mine at what Point in the Side of a Hill its Attraction will be the greatest. It occupies pages 1...14 of the volume: it was read Nov. 11, 1779. The memoir proposes to find at what point on the surface of a hill the horizontal component of the attraction of the hill is great- est. The problem was naturally suggested by the operations on the mountain Schehallien. Hutton supposes that the vertical section of his mountain is a triangle, and that the mountain extends to infinity in the hori- zontal direction on both sides of the point considered. Thus his problem may be stated in these words : find a point on a given face of a triangular prism of infinite length where the attraction of the prism resolved parallel to another given face is greatest. 472 MISCELLANEOUS INVESTIGATIONS BETWEEN 1761 AND 1780. Button's solution is wrong. I will briefly indicate the correct method. Let ABC be a section of the prism at right angles to its edges. Let P be any point on the side AB. Then we may divide the prism into two prisms, one corre- sponding to PEG, and the other to PAG, and estimate the attrac- tion of each separately. Suppose P the origin of polar coordinates, in the plane of the triangle; and take the initial line parallel to EG. The attraction of the infinite rod parallel to the edges of the 2 prism which corresponds to the polar element rdr dd is - rdr d9, where the density is taken to be unity. Consider first the prism corresponding to P